Geometrical Interpretation of Product of Vectors

Edited By Komal Miglani | Updated on Jul 02, 2025 07:44 PM IST

The cross( or vector) product of two vectors results in a vector. Based on this type of product for vectors, we have various applications in geometry, mechanics, and engineering. This application of vectors can be used to find the Area of a Parallelogram and the Area of a Triangle. In real life, we use vector( or cross ) products to find torque on a wrench, magnetic force on a moving electric charge, angular momentum of a rotating object, etc.

This Story also Contains

Geometrical Interpretation of Scalar Product
Geometrical Interpretation of Vector Product
1) Area of Parallelogram
2) Area of Triangle
Solved Examples Based on Geometrical Interpretation of Product of Vectors

Geometrical Interpretation of Product of Vectors

In this article, we will cover the concept of Geometrical Interpretation of Vector products. This topic falls under the broader category of Vector Algebra, which is a crucial chapter in Class 11 Mathematics. This is very important not only for board exams but also for competitive exams, which even include the Joint Entrance Examination Main and other entrance exams: SRM Joint Engineering Entrance, BITSAT, WBJEE, and BCECE. A total of fifteen questions have been asked on this topic in JEE Main from 2013 to 2023 including one in 2020, one in 2021, and three in 2023.

Geometrical Interpretation of Scalar Product

If $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ are two non-zero vectors, then their scalar product (or dot product) is denoted by $\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}$ and is defined as
$\overrightarrow{\mathbf{a}} \cdot \overrightarrow{\mathbf{b}}=|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}| \cos \theta . \quad(0 \leq \theta \leq \pi) \space{\text {where } \theta}$ is the angle between $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$

Let $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$ be two vectors represented by $O A$ and OB, respectively.
Draw $\mathrm{BL} \perp \mathrm{OA}$ and $\mathrm{AM} \perp \mathrm{OB}$.
From triangles $O B L$ and $O A M$ we have $O L=O B \cos \theta$ and $O M=O A \cos \theta$.
Here OL and OM are known as projections of $\overrightarrow{\mathbf{b}}$ on $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{a}}$ on $\overrightarrow{\mathbf{b}}$ respectively.

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Now, $\quad \begin{aligned} \vec{a} \cdot \vec{b} & =|\vec{a}||\vec{b}| \cos \theta \\ & =|\vec{a}|(O B \cos \theta) \\ & =|\vec{a}|(O L) \\ & =(\text { magnitude of } \vec{a})(\text { projection of } \vec{b} \text { on } \vec{a}) \\ \text { Again, } \quad \vec{a} \cdot \vec{b} & =|\vec{a}||\vec{b}| \cos \theta \\ & =|\vec{b}|(|\vec{a}| \cos \theta) \\ & =|\vec{b}|(O A \cos \theta) \\ & =|\vec{b}|(O M) \\ & =(\text { magnitude of } \vec{b})(\text { projection of } \vec{a} \text { on } \vec{b})\end{aligned}$

Thus. geometrically interpreted, the scalar product of two vectors is the product of the modulus of either vector and the projection of the other in its direction.

Thus,

Projection of $\overrightarrow{\mathbf{a}}$ on $\overrightarrow{\mathbf{b}}=\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}=\overrightarrow{\mathbf{a}} \cdot \frac{\vec{b}}{|\vec{b}|}=\overrightarrow{\mathbf{a}} \cdot \hat{\mathbf{b}}$
Projection of $\vec{b}$ on $\overrightarrow{\mathbf{a}}=\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|}=\vec{b} \cdot \frac{\vec{a}}{|\vec{a}|}=\vec{b} \cdot \hat{\mathbf{a}}$
What is Vector( or cross-product)?
The vector product of two nonzero vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, is denoted by $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}$ and defined as,

$
\vec{a} \times \vec{b}=|\vec{a}||\vec{b}| \sin \theta \hat{\mathbf{n}}
$

where $\theta$ is the angle between $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}, 0 \leq \theta \leq \pi$ and $\hat{\mathbf{n}}$ is a unit vector perpendicular to both $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, such that $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\hat{\mathbf{n}}$ form a right-hand system.

Geometrical Interpretation of Vector Product

The vector product of two nonzero vectors $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, is denoted by $\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}$ and defined as,

$
\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}=|\overrightarrow{\mathbf{a}}||\overrightarrow{\mathbf{b}}| \sin \theta \hat{\mathbf{n}}
$

where $\theta$ is the angle between $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}, 0 \leq \theta \leq \pi$ and $\hat{\mathrm{n}}$ is a unit vector perpendicular to both $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, such that $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\hat{\mathrm{n}}$ form a right-hand system.

1) Area of Parallelogram

The area of a parallelogram is the region covered by a parallelogram in the plane. If $\vec{a}$ and $\overrightarrow{\mathbf{b}}$, are two non-zero, non-parallel vectors representing two adjacent sides of the parallelogram then the modulus of cross product of the vector $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, represents the area of a parallelogram. If $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, are two non-zero, non-parallel vectors represented by $A D$ and $A B$ respectively and let $\theta$ be the angle between them.

$\begin{aligned} & \text { In } \triangle \mathrm{ADE}, \sin \theta=\frac{D E}{A D} \\ & \Rightarrow \quad D E=A D \sin \theta=|\overrightarrow{\mathbf{a}}| \sin \theta \\ & \text { Area of parallelogram } \mathrm{ABCD}=\mathrm{AB} \cdot \mathrm{DE} \\ & \text { Thus, } \\ & \text { Area of parallelogram } \mathrm{ABCD}=|\vec{b}||\vec{a}| \sin \theta=\mid \vec{a} \times \vec{b}\end{aligned}$

2) Area of Triangle

The region covered by a triangle in a plane is called the area of a Triangle. If $\vec{a}$ and $\overrightarrow{\mathbf{b}}$, are two non-zero, non-parallel vectors represented as the adjacent sides of a triangle then the area of a triangle is half of the modulus of the vector product of the vector $\vec{a}$ and $\vec{b}$. If $\overrightarrow{\mathbf{a}}$ and $\overrightarrow{\mathbf{b}}$, are two non-zero, non-parallel vectors represented as the adjacent sides of a triangle then its area is given as $\frac{1}{2}|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|$

The area of a triangle is $1 / 2$ (Base) $\times$ (Height)
From the figure,
Area of triangle $\mathrm{ABC}=\frac{1}{2} \mathrm{AB} \cdot \mathrm{CD}$
But $\mathrm{AB}=|\overrightarrow{\mathbf{b}}|$ (as given), and $\mathrm{CD}=|\overrightarrow{\mathbf{a}}| \sin \theta$
Thus, Area of triangle $\mathrm{ABC}=\frac{1}{2}|\overrightarrow{\mathbf{b}}||\overrightarrow{\mathbf{a}}| \sin \theta=\frac{1}{2}|\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}|$
NOTE:
1. The area of a parallelogram with diagonals $\overrightarrow{\mathbf{d}}_1$ and $\overrightarrow{\mathbf{d}}_2$ is $\frac{1}{2}\left|\overrightarrow{\mathbf{d}}_1 \times \overrightarrow{\mathbf{d}}_2\right|$.
2. The area of a plane quadrilateral $A B C D$ with AC and BD as diagonal is $\frac{1}{2}|\overrightarrow{\mathbf{A C}} \times \overrightarrow{\mathrm{BD}}|$.
3. If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are position vectors of a $\triangle A B C$, then its area is $\frac{1}{2}|(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})|$

Solved Examples Based on Geometrical Interpretation of Product of Vectors

Example 1: Let for a triangle ABC,

$\begin{aligned} & \overrightarrow{\mathrm{AB}}=-2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{CB}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}} \\ & \overrightarrow{\mathrm{CA}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\delta \hat{\mathrm{k}}\end{aligned}$

If $\delta>0$ and the area of the triangle ABC is $5 \sqrt{6}$, then $\overrightarrow{\mathrm{CB}} \cdot \overrightarrow{\mathrm{CA}}$ is equal to [JEE MAINS 2023]

Solution

$\begin{aligned} & \overrightarrow{\mathrm{CA}}+\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{CB}} \\ & \langle 4,3, \delta\rangle \cdot+\langle-2,1,3\rangle=\overrightarrow{\mathrm{CB}} \\ & \Rightarrow \overrightarrow{\mathrm{CB}}=\langle 2,4,3+\delta\rangle\end{aligned}$

$\begin{aligned} & \Rightarrow|\overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}|^2=600 \\ & \Rightarrow 5 \delta^2+30 \delta-275=0 \\ & \Rightarrow \mathrm{S}^2+6 \delta-55=0 \\ & \Rightarrow(\delta+11)(\delta-5)=0 \\ & \delta=5 \\ & \overrightarrow{\mathrm{CB}}=<2,3,8> \\ & \overrightarrow{\mathrm{CB}} \cdot \overrightarrow{\mathrm{CA}} \cdot=<2,4,8>\cdot<4,3,5> \\ & =8+12+40=60\end{aligned}$

Hence, the answer is 60

Example 2: Let $\vec{a}=4 \hat{i}+3 \hat{j}+5 \hat{k}$ and $\vec{\beta}=\hat{i}+2 \hat{j}-4 \hat{k}$, Let $\vec{\beta}_1$ be parallel to $\vec{a}$ and $\vec{\beta}_2$ be perpendicular to $\vec{a}$.If $\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2$, then the value of $5 \vec{\beta}_2 \cdot(\hat{i}+\hat{j}+\hat{k})$ is [JEE MAINS 2023]

Solution

$
\begin{aligned}
& \vec{\beta}_1=\frac{(\vec{\alpha} \cdot \vec{\beta})}{|\vec{\alpha}|} \hat{\alpha} \\
& =\left(\frac{4+6-20}{\sqrt{16+9+25}}\right) \frac{(4,3,5)}{\sqrt{50}} \\
& =\frac{-10}{50}(4,3,5) \\
& \vec{\beta}_1=\frac{(-4,-3,-5)}{5} \\
& \vec{\beta}_1+\vec{\beta}_2-=(1,2,-4) \\
& \beta_2=\left(1+\frac{4}{5}, 2+\frac{3}{5},-4+1\right) \\
& \beta_2=\left(\frac{9}{5}, \frac{13}{5},-3\right) \\
& \therefore 5 \beta_2=(9,13,-15) \\
& \therefore 5 \beta_2 \cdot(1,1,1)=9+13-15 \\
& =7
\end{aligned}
$
Hence, the answer is 7

Example 3: If $\mathrm{a}^{\prime}=\hat{\imath}+2 k, b=\hat{\imath}+\hat{\jmath}+k, \vec{c}=7 \hat{\imath}-3 \hat{\jmath}+4 k, \mathrm{r}^{\prime} \times \mathrm{b}+\mathrm{b} \times \mathrm{c}^{\prime}=0$ and $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}=0$. Then $\vec{r} \cdot \vec{c}$ is equal to
Solution

$
\begin{aligned}
& \vec{r} \times \vec{b}+\vec{b} \times \vec{c}=0 \\
& \Rightarrow \vec{r} \times \vec{b}-\vec{c} \times \vec{b}=0 \\
& \Rightarrow(\vec{r}-\vec{c}) \times \vec{b}=0 \\
& \vec{r}-\vec{c} \| \vec{b} \\
& \vec{r}-\vec{c}=\lambda \vec{b} \\
& \vec{r}=\lambda \vec{b}+\vec{c} \\
& =\lambda(i+j+k)+(7 i-3 j+4 k) \\
& =\mathrm{i}(\lambda+7)+j(\lambda-3)+\mathrm{k}(\lambda+4) \\
& \vec{r} \cdot \vec{a}=0 \\
& \Rightarrow(7+\lambda)+2(\lambda+4)=0 \\
& \Rightarrow 3 \lambda=-15 \Rightarrow \lambda=-5 \\
& \therefore \vec{r}=2 \mathrm{i}-8 \mathrm{j}-\mathrm{k} \\
& \overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{c}}=(2 \mathrm{i}-8 \mathrm{j}-\mathrm{k}) \cdot(7 \mathrm{i}-3 \mathrm{j}+4 \mathrm{k}) \\
& =14+24-4=34
\end{aligned}
$
Hence, the answer is 34

Example 4: Let $\vec{a}=\hat{i}+2 \hat{j}-\hat{k}, \vec{b}=\hat{i}-2 \hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}$ is a vector such that $\vec{r} \times \vec{a}=\vec{c} \times \vec{a}$ and $\vec{r} \cdot \vec{b}=0$, then $\vec{r} \cdot \vec{a}$ is equal to
Solution

$
\begin{aligned}
& (\vec{r}-\vec{c}) \times \vec{a}=0 \\
& \Rightarrow \vec{r}=\vec{c}+\lambda \vec{a}
\end{aligned}
$
Now, $0=\vec{b} \cdot \vec{c}+\lambda \vec{a} \cdot \vec{b}$

$
\Rightarrow \lambda=\frac{-\vec{b} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}=-\frac{2}{-1}=2
$
So, $\vec{r} \cdot \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}+2 \mathrm{a}^2=12$
Hence, the answer is 12

Example 5: The area (in sq. units) of the parallelogram whose diagonals are along the vectors $8 \hat{i}-6 \hat{j}$ and $3 \hat{i}+4 \hat{j}-12 \hat{k}$,
[JEE MAINS 2017]
Solution: Area of parallelogram $=\frac{1}{2}|\vec{a} \times \vec{b}|$
where $\vec{a}$ and $\vec{b}$ are diagonals

$
\begin{aligned}
& =\frac{1}{2}\left|\begin{array}{ccc}
\hat{i} & \hat{j} & \hat{k} \\
8 & -6 & 0 \\
3 & 4 & -12
\end{array}\right| \\
& =\left|\frac{1}{2}(72 \hat{i}+96 \hat{j}+50 \hat{k})\right| \\
& =|36 \hat{i}+48 \hat{j}+25 \hat{k}| \\
& \text { magnitude }=\sqrt{36^2+48^2+25^2} \\
& =65
\end{aligned}
$
Hence, the answer is 65

Frequently Asked Questions (FAQs)

1. What is the geometrical interpretation of the cross product of two vectors?

The cross product of two vectors geometrically represents a vector perpendicular to both original vectors, with a magnitude equal to the area of the parallelogram formed by the two vectors. The direction of the resulting vector follows the right-hand rule.

2. How does the magnitude of the cross product relate to the angle between vectors?

The magnitude of the cross product is proportional to the sine of the angle between the two vectors. Geometrically, this means the cross product is largest when the vectors are perpendicular and zero when they are parallel or antiparallel.

3. What is the geometrical significance of the scalar triple product?

The scalar triple product of three vectors geometrically represents the volume of the parallelepiped formed by these vectors. It can be visualized as the product of the magnitude of one vector and the area of the parallelogram formed by the other two vectors.

4. What does the direction of the cross product vector signify geometrically?

The direction of the cross product vector is perpendicular to the plane containing the two original vectors. It follows the right-hand rule, which geometrically determines the orientation of the resulting vector in three-dimensional space.

5. How can the cross product be used to determine if two vectors are parallel?

If the cross product of two vectors is zero, it geometrically indicates that the vectors are parallel or antiparallel. This is because parallel vectors do not form a plane, resulting in a cross product with no magnitude or direction.

6. What is the geometrical interpretation of the parallelogram law of vector addition?

The parallelogram law geometrically represents vector addition by forming a parallelogram with the two vectors as adjacent sides. The diagonal of this parallelogram represents the sum vector, both in magnitude and direction.

7. How does the cross product help in calculating the area of a parallelogram?

The magnitude of the cross product of two vectors is equal to the area of the parallelogram formed by these vectors. Geometrically, this represents the relationship between the perpendicular component of one vector relative to another and the area they enclose.

8. How does the Cauchy-Schwarz inequality relate to the geometrical interpretation of vectors?

The Cauchy-Schwarz inequality (|a · b| ≤ |a||b|) geometrically states that the magnitude of the dot product of two vectors is always less than or equal to the product of their magnitudes. This is visually represented by the fact that the projection of one vector onto another is always shorter than or equal to its full length.

9. What is the geometrical significance of the vector projection formula?

The vector projection formula geometrically represents the shadow cast by one vector onto another. It gives the component of one vector that is parallel to another vector, effectively breaking down a vector into its parallel and perpendicular components relative to another vector.

10. What is the geometrical significance of the vector triple product expansion?

The vector triple product expansion a × (b × c) = (a · c)b - (a · b)c geometrically represents the decomposition of the rejection of vector a from the plane of b and c into components parallel to b and c.

11. What is the geometrical interpretation of the dot product of two vectors?

The dot product of two vectors geometrically represents the product of the magnitude of one vector and the projection of the other vector onto it. It can be visualized as the length of the shadow one vector casts on the other when light shines perpendicular to it.

12. What does a positive dot product indicate geometrically?

A positive dot product indicates that the two vectors point in generally the same direction, forming an acute angle (less than 90°) between them. Geometrically, this means the projection of one vector onto the other is in the same direction as the vector being projected upon.

13. How is the dot product related to vector magnitudes?

The dot product of two vectors is equal to the product of their magnitudes multiplied by the cosine of the angle between them. Geometrically, this means the dot product is influenced by both the lengths of the vectors and how closely aligned they are in direction.

14. What is the geometrical significance of a zero dot product?

A zero dot product geometrically signifies that the two vectors are perpendicular (orthogonal) to each other. This means that the projection of one vector onto the other results in a point, indicating no component of one vector in the direction of the other.

15. How does the dot product help in determining vector orientation?

The dot product helps determine vector orientation by indicating whether two vectors form an acute angle (positive dot product), obtuse angle (negative dot product), or right angle (zero dot product). This geometric interpretation aids in understanding the relative directions of vectors in space.

16. How does the angle between two vectors affect their dot product?

The angle between two vectors directly influences their dot product. As the angle increases from 0° to 180°, the dot product decreases from its maximum positive value to its minimum negative value. When the vectors are perpendicular (90°), their dot product is zero.

17. How does the dot product relate to work done in physics?

In physics, work is calculated as the dot product of force and displacement vectors. Geometrically, this represents the component of force acting in the direction of displacement, multiplied by the magnitude of displacement.

18. What is the geometrical interpretation of the scalar projection of one vector onto another?

The scalar projection of vector a onto vector b is the length of the shadow cast by a onto b when light shines perpendicular to b. Geometrically, it represents the component of a in the direction of b, expressed as a scalar quantity.

19. What is the geometrical significance of the vector equation of a plane?

The vector equation of a plane r · n = d, where n is normal to the plane, geometrically represents that the dot product of any point r in the plane with the normal vector n is constant. This constant d is related to the distance of the plane from the origin.

20. How does the cross product help in calculating the moment of inertia?

The moment of inertia involves the cross product r × (r × ω), where r is the position vector and ω is the angular velocity. Geometrically, this represents how the distribution of mass affects rotational inertia based on its perpendicular distance from the axis of rotation.

21. How does the scalar triple product help in determining coplanarity of vectors?

If the scalar triple product of three vectors is zero, it geometrically indicates that the vectors are coplanar (lie in the same plane). This is because coplanar vectors form a parallelepiped with zero volume.

22. What is the geometrical interpretation of the vector triple product?

The vector triple product a × (b × c) can be geometrically interpreted as the rejection of vector a from the plane formed by vectors b and c. It lies in this plane and is perpendicular to the cross product b × c.

23. How does the distributive property of dot product over addition relate geometrically?

Geometrically, the distributive property of dot product over addition (a · (b + c) = a · b + a · c) represents that the projection of a vector onto the sum of two vectors is equal to the sum of its projections onto each individual vector.

24. What is the geometrical meaning of the magnitude of a vector in terms of dot product?

The magnitude of a vector can be geometrically interpreted as the square root of the dot product of the vector with itself. This represents the length of the vector in Euclidean space.

25. How does the dot product help in finding the angle between two vectors geometrically?

The dot product allows us to find the angle between two vectors through the formula cos θ = (a · b) / (|a||b|). Geometrically, this represents the relationship between the projection of one vector onto another and their magnitudes.

26. What is the geometrical interpretation of the cross product in terms of torque?

In physics, torque is represented by the cross product of force and position vectors. Geometrically, this indicates that torque is a vector perpendicular to both the force and the lever arm, with magnitude proportional to the sine of the angle between them.

27. How does the scalar triple product relate to the volume of a parallelepiped?

The scalar triple product a · (b × c) geometrically represents the volume of the parallelepiped formed by vectors a, b, and c. It can be visualized as the product of the base area (formed by b × c) and the height (projection of a onto b × c).

28. What is the geometrical significance of the vector component formula?

The vector component formula represents the breakdown of a vector into its components along different directions. Geometrically, it shows how a vector can be expressed as the sum of its projections onto different axes or directions.

29. How does the cross product help in determining the normal vector to a plane?

The cross product of two non-parallel vectors lying in a plane produces a vector perpendicular to that plane. Geometrically, this provides a method to find the normal vector to a plane given two vectors in the plane.

30. What is the geometrical interpretation of the dot product of perpendicular vectors?

The dot product of perpendicular vectors is always zero. Geometrically, this means that the projection of one vector onto the other results in a point, indicating no component of one vector in the direction of the other.

31. How does the magnitude of the cross product relate to the sine of the angle between vectors?

The magnitude of the cross product |a × b| = |a||b|sin θ geometrically represents the area of the parallelogram formed by the two vectors. This area is maximum when the vectors are perpendicular (sin θ = 1) and zero when they are parallel (sin θ = 0).

32. What is the geometrical significance of the vector rejection?

The vector rejection represents the component of one vector that is perpendicular to another vector. Geometrically, it's the difference between the original vector and its projection onto the other vector, forming a right triangle with the original vector and its projection.

33. How does the dot product help in determining orthogonality of vectors?

Two vectors are orthogonal (perpendicular) if their dot product is zero. Geometrically, this means that neither vector has a component in the direction of the other, forming a right angle between them.

34. How does the cross product relate to the area of a triangle formed by two vectors?

The magnitude of the cross product of two vectors is twice the area of the triangle formed by these vectors. Geometrically, this relationship arises because the cross product gives the area of a parallelogram, which is twice the area of the triangle formed by the same vectors.

35. What is the geometrical significance of the vector triple product identity?

The vector triple product identity a × (b × c) = (a · c)b - (a · b)c geometrically represents the decomposition of the rejection of vector a from the plane of b and c. It shows how this rejection can be expressed in terms of components parallel to b and c.

36. How does the dot product help in finding the projection matrix?

The projection matrix, derived from the dot product, geometrically represents the operation of projecting vectors onto a subspace. It allows us to find the component of any vector that lies in a particular subspace defined by other vectors.

37. What is the geometrical interpretation of the cross product of parallel vectors?

The cross product of parallel vectors is always the zero vector. Geometrically, this means that parallel vectors do not form a plane, resulting in no perpendicular vector and thus no area enclosed between them.

38. How does the scalar triple product change with cyclic permutation of vectors?

The scalar triple product a · (b × c) remains unchanged under cyclic permutation of vectors (a · (b × c) = b · (c × a) = c · (a × b)). Geometrically, this represents that the volume of the parallelepiped remains the same regardless of which vector is considered as the height.

39. How does the cross product help in determining the direction of magnetic force?

In physics, the magnetic force on a moving charged particle is given by the cross product of its velocity and the magnetic field vectors. Geometrically, this means the force is perpendicular to both the velocity and the magnetic field, following the right-hand rule.

40. What is the geometrical interpretation of the vector form of Lagrange's identity?

Lagrange's identity |a × b|² = |a|²|b|² - (a · b)² geometrically relates the area of the parallelogram formed by two vectors (left side) to the difference between the product of their squared magnitudes and the square of their dot product (right side).

41. How does the dot product relate to the law of cosines in trigonometry?

The dot product formula a · b = |a||b|cos θ is a generalization of the law of cosines. Geometrically, it relates the angle between two vectors to their magnitudes and the projection of one onto the other, similar to how the law of cosines relates the sides of a triangle to its angles.

42. What is the geometrical significance of the vector triple product being zero?

If the vector triple product a × (b × c) is zero, it geometrically indicates that the three vectors are coplanar. This means that vector a lies in the plane formed by vectors b and c, or that all three vectors are parallel or zero.

43. How does the cross product help in determining the chirality of a coordinate system?

The cross product follows the right-hand rule, which geometrically determines the orientation of a coordinate system. If i × j = k in a coordinate system, it's right-handed; if i × j = -k, it's left-handed.

44. What is the geometrical interpretation of the dot product in terms of work and force?

In physics, the dot product of force and displacement vectors represents work done. Geometrically, this can be interpreted as the product of the magnitude of the force and the displacement in the direction of the force.

45. How does the scalar triple product relate to the independence of vectors?

If the scalar triple product of three vectors is non-zero, it geometrically indicates that the vectors are linearly independent and span a three-dimensional space. A zero scalar triple product means the vectors are coplanar or linearly dependent.

46. What is the geometrical significance of the vector rejection being perpendicular to the projection?

The vector rejection of a onto b is always perpendicular to the projection of a onto b. Geometrically, this forms a right triangle where the original vector a is the hypotenuse, and the projection and rejection are the other two sides.

47. What is the geometrical interpretation of the vector form of the distributive property of cross products?

The distributive property a × (b + c) = a × b + a × c geometrically means that the area of the parallelogram formed by a and the sum of b and c is equal to the sum of the areas of parallelograms formed by a with b and a with c separately.

48. How does the dot product relate to the concept of orthogonal projections in linear algebra?

The dot product is fundamental in calculating orthogonal projections. Geometrically, the formula for the orthogonal projection of a onto b, (a · b / |b|²)b, represents the component of a that is parallel to b, scaled by the unit vector in the direction of b.

49. What is the geometrical significance of the cross product being anticommutative?

The anticommutative property of cross products (a × b = -b × a) geometrically means that changing the order of vectors in a cross product reverses the direction of the resulting vector while maintaining its magnitude. This is related to the right-hand rule for determining cross product direction.

50. How does the scalar triple product help in determining the handedness of a coordinate system?

The scalar triple product i · (j × k) is positive for a right-handed coordinate system and negative for a left-handed one. Geometrically, this represents whether the shortest rotation from j to k is in the same direction as i (right-handed) or opposite to i (left-handed).