Vector Triple Product

What is a Vector Triple Product?

The vector triple product is an important operation in vector algebra that involves three vectors. It is obtained when the cross product of two vectors is again crossed with a third vector. Unlike the scalar triple product, the result of a vector triple product is a vector. This concept is widely used in mathematics, physics, engineering, and three-dimensional geometry to simplify complex vector expressions and analyze spatial relationships.

Vector Triple Product Meaning in Simple Words

A vector triple product can be thought of as a two-step vector operation. First, two vectors are combined using a cross product, and then the resulting vector is crossed with another vector. This operation helps express complicated vector relationships in a simpler form.

Definition of Vector Triple Product

The vector triple product of three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is defined as

$\vec{a}\times(\vec{b}\times\vec{c})$ or

$(\vec{a}\times\vec{b})\times\vec{c}$

Both expressions involve three vectors but generally produce different results because the cross product is not associative.

Why the Vector Triple Product is Important

The vector triple product helps simplify lengthy vector calculations and provides important identities used in higher mathematics and physics. It is particularly useful when solving problems involving forces, moments, electromagnetic fields, and coordinate transformations.

Real-Life Applications of Vector Triple Product

The vector triple product is used in:

• Mechanics and force systems

• Electromagnetism

• Fluid dynamics

• Robotics

• Aerospace engineering

• Computer graphics and animation

• Three-dimensional modeling

Basics of Vector Algebra

Before studying vector triple products, it is important to understand the fundamental concepts of vector algebra.

What is a Vector?

A vector is a quantity that has both magnitude and direction.

Examples include:

• Velocity

• Force

• Acceleration

• Displacement

Vectors are usually represented by bold letters such as $\vec{a}$, $\vec{b}$, and $\vec{c}$.

Scalar and Vector Quantities

A scalar quantity has only magnitude.

Examples:

• Mass

• Temperature

• Time

• Distance

A vector quantity has both magnitude and direction.

Examples:

• Velocity

• Force

• Momentum

• Electric field

Dot Product of Two Vectors

The dot product of vectors $\vec{a}$ and $\vec{b}$ is given by $\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta$

The result is a scalar quantity.

The dot product is commonly used to determine angles between vectors and projections.

Cross Product of Two Vectors

The cross product of vectors $\vec{a}$ and $\vec{b}$ is

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

where $\hat{n}$ is a unit vector perpendicular to both vectors.

The result is a vector quantity.

Vector Triple Product Formula

The vector triple product can be simplified using a famous vector identity known as the BAC-CAB rule.

Standard Vector Triple Product Formula

The standard formula is

$\vec{a}\times(\vec{b}\times\vec{c})$

Using vector identities, it can be simplified considerably.

BAC-CAB Rule

The most important identity for vector triple products is

$\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}$

This identity is commonly called the BAC-CAB rule because the vectors appear in the order:

"BAC minus CAB."

Meaning of Terms in the Formula

In $\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}$

• $\vec{a},\vec{b},\vec{c}$ are vectors.

• $\vec{a}\cdot\vec{c}$ is a scalar.

• $\vec{a}\cdot\vec{b}$ is a scalar.

• The final result is a vector.

Mathematical Representation

The vector triple product can be represented as

$\vec{a}\times(\vec{b}\times\vec{c})$ or

$(\vec{a}\times\vec{b})\times\vec{c}$

These two expressions are generally not equal.

Derivation of the Vector Triple Product Formula

The vector triple product identity is one of the most important results in vector algebra. It helps convert a complicated cross product involving three vectors into a simpler linear combination of vectors.

We want to derive:

This identity is commonly known as the BAC-CAB Rule.

Step 1: Consider the Vector Triple Product

Let

$\vec{R}=\vec{a}\times(\vec{b}\times\vec{c})$

The vector $\vec{b}\times\vec{c}$ is perpendicular to both $\vec{b}$ and $\vec{c}$.

Since $\vec{R}$ is perpendicular to $(\vec{b}\times\vec{c})$, it must lie in the plane containing $\vec{b}$ and $\vec{c}$.

Therefore, $\vec{R}$ can be expressed as

$\vec{R}=m\vec{b}+n\vec{c}$

where $m$ and $n$ are scalars.

Step 2: Take Dot Product with $\vec{a}$

Taking the dot product of both sides with $\vec{a}$,

$\vec{a}\cdot\vec{R}=\vec{a}\cdot(m\vec{b}+n\vec{c})$

$\vec{a}\cdot\vec{R}=m(\vec{a}\cdot\vec{b})+n(\vec{a}\cdot\vec{c})$

Since

$\vec{R}=\vec{a}\times(\vec{b}\times\vec{c})$

and a vector is always perpendicular to its cross product,

$\vec{a}\cdot\vec{R}=0$

Hence,

$m(\vec{a}\cdot\vec{b})+n(\vec{a}\cdot\vec{c})=0$

Step 3: Take Dot Product with $\vec{b}$

Taking the dot product of

$\vec{R}=m\vec{b}+n\vec{c}$

with $\vec{b}$,

$\vec{R}\cdot\vec{b}=m(\vec{b}\cdot\vec{b})+n(\vec{b}\cdot\vec{c})$

Now,

$\vec{R}\cdot\vec{b}=[\vec{a}\times(\vec{b}\times\vec{c})]\cdot\vec{b}$

Using scalar triple product properties,

$\vec{R}\cdot\vec{b}=(\vec{b}\times\vec{c})\cdot(\vec{b}\times\vec{a})$

After simplification,

$\vec{R}\cdot\vec{b}=(\vec{a}\cdot\vec{c})|\vec{b}|^2-(\vec{a}\cdot\vec{b})(\vec{b}\cdot\vec{c})$

Thus,

$m|\vec{b}|^2+n(\vec{b}\cdot\vec{c})=(\vec{a}\cdot\vec{c})|\vec{b}|^2-(\vec{a}\cdot\vec{b})(\vec{b}\cdot\vec{c})$

Step 4: Compare Coefficients

The above relation is satisfied when

$m=(\vec{a}\cdot\vec{c})$ and

$n=-(\vec{a}\cdot\vec{b})$

Substituting these values into

$\vec{R}=m\vec{b}+n\vec{c}$

gives $\vec{R}=(\vec{a}\cdot\vec{c})\vec{b} = (\vec{a}\cdot\vec{b})\vec{c}$

Step 5: Final Result

Since

$\vec{R}=\vec{a}\times(\vec{b}\times\vec{c})$

we obtain $\boxed{\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}}$

This is the Vector Triple Product Formula or BAC-CAB Identity.

Why is it Called BAC-CAB?

Observe the final expression:

$(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}$

The vectors appear in the order:

BAC − CAB

which gives the famous memory trick:

BAC minus CAB

Important Observation

The result contains only $\vec{b}$ and $\vec{c}$.

This means that

$\vec{a}\times(\vec{b}\times\vec{c})$

always lies in the plane formed by $\vec{b}$ and $\vec{c}$, which is one of the most important geometric properties of the vector triple product.

Verification Using Cartesian Components

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$

$\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$

$\vec{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$

Expanding both sides yields identical expressions, proving the identity.

Properties of Vector Triple Product

Several important properties make vector triple products useful in vector algebra.

Non-Associative Nature

Cross products are not associative.

Therefore,

$\vec{a}\times(\vec{b}\times\vec{c})\neq(\vec{a}\times\vec{b})\times\vec{c}$

This is one of the most important facts to remember.

Coplanarity Property

The vector

$\vec{a}\times(\vec{b}\times\vec{c})$

lies in the plane containing $\vec{b}$ and $\vec{c}$.

Linear Combination Property

The BAC-CAB identity shows that the result is a linear combination of $\vec{b}$ and $\vec{c}$.

Orthogonality Relationships

The resulting vector remains perpendicular to the vector generated by the inner cross product.

This property is useful in geometric proofs and mechanics.

Geometrical Interpretation of Vector Triple Product

The vector triple product has a meaningful geometric interpretation.

Understanding the Resultant Vector

The resulting vector is always confined to the plane formed by two of the participating vectors.

Vector Triple Product and Planes

The BAC-CAB identity demonstrates that the resultant vector belongs to the plane spanned by $\vec{b}$ and $\vec{c}$.

Direction of the Resultant Vector

The direction depends on the magnitudes and relative orientations of the vectors involved.

Geometric Visualization in Three Dimensions

In three-dimensional space, vector triple products help describe projections, rotations, and vector decompositions.

Types of Triple Products in Vector Algebra

Vector algebra contains two major types of triple products.

Vector Triple Product

The vector triple product is

$\vec{a}\times(\vec{b}\times\vec{c})$

and produces a vector.

Scalar Triple Product

The scalar triple product is

$\vec{a}\cdot(\vec{b}\times\vec{c})$

and produces a scalar.

Difference Between Scalar and Vector Triple Products

The scalar triple product gives volume information, while the vector triple product gives directional information.

Applications of Each Type

Scalar triple products are used in volume calculations.

Vector triple products are used in force systems, projections, and vector simplifications.

Vector Triple Product in Component Form

Component methods are often used to simplify calculations.

Using Cartesian Coordinates

Vectors are expressed using $\hat{i}$, $\hat{j}$, and $\hat{k}$ components.

Expansion in Terms of Unit Vectors

The vectors are written as

$\vec{a}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$

and similarly for other vectors.

Simplifying Component Calculations

Component methods eliminate geometric ambiguity and make calculations systematic.

Common Computational Techniques

Common methods include:

• Determinant expansion

• BAC-CAB identity

• Unit vector decomposition

• Matrix methods

Applications of Vector Triple Product

Vector triple products appear in numerous scientific fields.

Vector Triple Product and Vector Projections

Vector triple products are closely related to projections.

Relationship with Projection of Vectors

The BAC-CAB rule expresses vectors using projections onto other vectors.

Resolving Vectors into Components

Vector decomposition is often simplified using triple product identities.

Projection Along a Given Direction

The identity helps determine vector components parallel to specified directions.

Geometric Significance

It provides a mathematical way to describe how vectors influence each other geometrically.

Vector Triple Product Identities

Several useful identities are associated with vector triple products.

First Vector Triple Product Identity

$\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{a}\cdot\vec{b})\vec{c}$

Second Vector Triple Product Identity

$(\vec{a}\times\vec{b})\times\vec{c}=(\vec{a}\cdot\vec{c})\vec{b}-(\vec{b}\cdot\vec{c})\vec{a}$

Related Vector Algebra Identities

Important identities include:

$\vec{a}\cdot(\vec{b}\times\vec{c})=\vec{b}\cdot(\vec{c}\times\vec{a})$ and $\vec{a}\times\vec{a}=0$

Important Results to Remember

• Cross products are not associative.

• BAC-CAB is the key identity.

• The result is always a vector.

• The resultant vector lies in a plane.

Difference Between Cross Product and Vector Triple Product

Although related, these operations are different.

Definition Comparison

Cross product involves two vectors.

Vector triple product involves three vectors.

Formula Comparison

Cross Product:

$\vec{a}\times\vec{b}$

Vector Triple Product:

$\vec{a}\times(\vec{b}\times\vec{c})$

Resultant Quantity Comparison

Both produce vectors, but vector triple products simplify into combinations of vectors.

Comparison Table

Common Mistakes in Vector Triple Product

Students frequently make several avoidable errors.

Assuming Associative Property

The most common mistake is assuming

$\vec{a}\times(\vec{b}\times\vec{c})=(\vec{a}\times\vec{b})\times\vec{c}$

which is false.

Incorrect Order of Vectors

Changing the order changes the result.

Vector order must always be preserved.

Sign Errors in Expansion

Students often forget the negative sign in the BAC-CAB identity.

Confusing Scalar and Vector Triple Products

Always check whether the operation begins with a dot product or a cross product.

Important Results

The following formulas are the most important for revision.

BAC-CAB Formula Table

Vector Product Identities Table

Key Properties at a Glance

• Result is always a vector.

• BAC-CAB rule simplifies the expression.

• Cross products are not associative.

• Resultant vector lies in the plane of $\vec{b}$ and $\vec{c}$.

• Widely used in physics, engineering, and three-dimensional geometry.

Best Books for Vector Triple Product

Vector triple products are an important part of vector algebra and three-dimensional geometry. The following books explain vector operations, vector identities, and applications in physics and engineering.

Shortcut Tips and Tricks for Vector Triple Product

Vector triple product problems can often be simplified using standard identities and geometric interpretations.

Important Formula Table

The following formulas are frequently used in vector triple product problems.

Solved Examples Based on Vector Triple Product

Example 1: If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\hat{c}$ such that $\vec{a}=\alpha \vec{b}-\hat{n}(a \neq 0)$ and $\vec{b} \cdot \vec{c}=12$, then $|\vec{c} \times(\vec{a} \times \vec{b})|_{\text {is equal to: }}$
[JEE MAINS 2023]

Solution:

$
\begin{aligned}
& \overrightarrow{\mathrm{a}}=\alpha \overrightarrow{\mathrm{b}}-\hat{\mathrm{n}}, \vec{b} \cdot \overrightarrow{\mathrm{c}}=12 \\
& \overrightarrow{\mathrm{c}} \times(\vec{a} \times \vec{b})=(\vec{c} \cdot \vec{b}) \vec{a}-(\vec{c} \cdot \vec{a}) \vec{b} \\
& \overrightarrow{\mathrm{c}} \times(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})=12 \vec{a}-(\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{b}}) \\
& \because \overrightarrow{\mathrm{a}}=\alpha \overrightarrow{\mathrm{b}}-\mathrm{n}
\end{aligned}
$
$
\vec{c} \cdot \vec{a}=\alpha \overrightarrow{\mathrm{c}} \cdot \vec{b}-\vec{c} \cdot \mathrm{n}
$
$
\vec{c} \cdot \vec{a}=12 \alpha
$
$
\begin{aligned}
& \vec{c} \times(\vec{a} \times \vec{b})=12 \vec{a}-12 \alpha \vec{b} \\
& |\vec{c} \times(\vec{a} \times \vec{b})|=12|\vec{a}-\alpha \vec{b}| \quad[\because \vec{a}-\alpha \vec{b}=-n \text { then }|\vec{a}-\alpha \vec{b}|=1] \\
& \Rightarrow|\vec{c} \times(\vec{a} \times \vec{b})|=12
\end{aligned}
$
$
|\overrightarrow{\mathrm{c}} \times(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})|=12
$
Hence, the answer is 12

Example 2: Let $\lambda \in \mathbb{R}, \vec{a}=\lambda \hat{\imath}+2 \hat{\jmath}-3 \hat{k}, \vec{b}=\hat{\imath}-\lambda \hat{\jmath}+2 \hat{k}{ }_{\operatorname{If}}((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b})=8 \hat{\imath}-40 \hat{\jmath}-24 \hat{k}$, then $|\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|^2$ is equal to

Solution:

$
\begin{aligned}
& ((\vec{a}+\vec{b}) \times(\vec{a} \times \vec{b}) \times(\vec{a}-\vec{b})-=8 \hat{i}-40 \hat{j}-24 \hat{k} \\
& \Rightarrow(\vec{a} \times(\vec{a} \times \vec{b})+\vec{b} \times(\vec{a} \times \vec{b})) \times(\vec{a}-\vec{b}) \\
& \Rightarrow((\vec{a} \cdot \vec{b}) \vec{a}-(\vec{a} \cdot \vec{a}) \vec{b}+(\vec{b} \cdot \vec{b}) \vec{a}-(\vec{b} \cdot \vec{a}) \vec{b}) \times(\vec{a}-\vec{b}) \\
& \Rightarrow 0-(\vec{a} \cdot \vec{b})(\vec{a} \times \vec{b})-a^2(\vec{b} \times \vec{a})+0-\mathrm{b}^2(\vec{a} \times \vec{b})-(\vec{a} \cdot \vec{b}) \vec{b} \times \vec{a}=8 \hat{\mathrm{i}}-40 \hat{\mathrm{j}}-24 \hat{\mathrm{k}} \\
& \Rightarrow\left(\mathrm{a}^2-\mathrm{b}^2\right)(\vec{a} \times \overrightarrow{\mathrm{b}})=8 \hat{\mathrm{i}}-40 \hat{\mathrm{j}}-24 \hat{\mathrm{k}} \\
& \left(\left(\lambda^2+4+9\right)-\left(1+\lambda^2+4\right)\right)(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \\
& 8(\vec{a} \times \vec{b})=8(\hat{\mathrm{i}}-5 \hat{j}-3 \hat{k}) \\
& \hat{\mathrm{i}}(4-3 \lambda)-\hat{\mathrm{j}}(2 \lambda+3)+\hat{\mathrm{k}}\left(-\lambda^2-2\right)=\hat{\mathrm{i}}-5 \hat{\mathrm{j}}-3 \hat{\mathrm{k}} \\
& \Rightarrow 4-3 \lambda=1 \quad 2 \lambda+3=5 \quad-\lambda^2-2=-3 \\
& 3 \lambda=3 \\
& \lambda=1 \\
& \lambda \\
& \begin{array}{l}
\lambda^2=1
\end{array} \\
& \begin{array}{ll}
\lambda(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})|=|(\vec{a}+\vec{b}) \times\left.(\vec{a}-\vec{b})\right|^2 \\
\Rightarrow|-\vec{a} \times \vec{b}+\vec{b} \times \vec{a}|^2=|2(\vec{a} \times \vec{b})|^2=4(1+25+9)=140
\end{array}
\end{aligned}
$
Hence, the answer is 140

Example 3: Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b} \cdot \vec{c}=0$ and $\vec{a} \times(\vec{b} \times \vec{c})=\frac{b-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b}$, then $(\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})_{\text {is equal to }}$.
[JEE MAINS 2023]

Solution:

We are given $\vec{b}\cdot\vec{c}=0$ and $\vec{a}\times(\vec{b}\times\vec{c})=\dfrac{\vec{b}-\vec{c}}{2}$

Use the vector triple product identity

$\vec{a}\times(\vec{b}\times\vec{c})=\vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b})$

So, $\vec{b}(\vec{a}\cdot\vec{c})-\vec{c}(\vec{a}\cdot\vec{b})=\dfrac{\vec{b}-\vec{c}}{2}$

Compare coefficients of $\vec{b}$ and $\vec{c}$ on both sides.

For $\vec{b}$: $\vec{a}\cdot\vec{c}=\dfrac{1}{2}$

For $\vec{c}$: $-(\vec{a}\cdot\vec{b})=-\dfrac{1}{2}$

So, $\vec{a}\cdot\vec{b}=\dfrac{1}{2}$

Now we are given $\vec{b}\cdot\vec{d}=\vec{a}\cdot\vec{b}=\dfrac{1}{2}$

We need to find $(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})$

Use the identity

$(\vec{p}\times\vec{q})\cdot(\vec{r}\times\vec{s})=(\vec{p}\cdot\vec{r})(\vec{q}\cdot\vec{s})-(\vec{p}\cdot\vec{s})(\vec{q}\cdot\vec{r})$

So, $(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})=(\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d})-(\vec{a}\cdot\vec{d})(\vec{b}\cdot\vec{c})$

But $\vec{b}\cdot\vec{c}=0$, hence the second term vanishes.

So, $(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})=(\vec{a}\cdot\vec{c})(\vec{b}\cdot\vec{d})$

Substitute known values:

$\vec{a}\cdot\vec{c}=\dfrac{1}{2}$

$\vec{b}\cdot\vec{d}=\dfrac{1}{2}$

Therefore, $(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})=\dfrac{1}{2}\cdot\dfrac{1}{2}$

$=\dfrac{1}{4}$

Hence, the answer is $\frac{1}{4}$.

Example 4: Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have the same magnitude. If a vector $\vec{r}$ satisfies $\vec{a} \times\{(\vec{r}-b) \times \vec{a}\}+b \times\{(\vec{r}-\vec{c}) \times b\}+\vec{c} \times\{(\vec{r}-\vec{a}) \times \vec{c}\}=0$,then $\vec{r}$ is equal to:

Solution:

$|\vec{a}|=|\vec{b}|=|\vec{c}|$ and $\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{c}=\vec{c} \cdot \vec{a}=0$
Let $\vec{r}=x \vec{a}+y \vec{b}+z \vec{c}$
where $\vec{r} \cdot \vec{a}=x|\vec{a}|^2, \vec{r} \cdot \vec{b}=y|\vec{b}|^2, \vec{r} \cdot \vec{c}=z|\vec{c}|^2$
Give expression is

$
\begin{aligned}
& (\vec{a} \times(\vec{r} \times \vec{a}))-(\vec{a} \times(\vec{b} \times \vec{a}))+\vec{b} \times(\vec{r} \times \vec{b})-\vec{b} \times(\vec{c} \times \vec{b})+ \\
& \vec{c} \times(\vec{r} \times \vec{c})-(\vec{c} \times(\vec{a} \times c))=0 \\
& \Rightarrow(\vec{a} \cdot \vec{r}) \vec{a}-|\vec{a}|^2 \vec{r}-(\vec{a} \cdot \vec{b}) \vec{a}+|\vec{a}|^2 \vec{b}+(\vec{b} \cdot \vec{r}) \vec{b}-|\vec{b}|^2 \vec{r}- \\
& (\vec{b} \cdot \vec{c}) \vec{b}+|\vec{b}|^2 \vec{c}+(\vec{c} \cdot \vec{r}) \vec{c}-|\vec{c}|^2 \vec{r}-(\vec{c} \cdot \vec{a}) \vec{a}+|\vec{c}|^2 \vec{a}=0 \\
& \Rightarrow x|\vec{a}|^2 \vec{a}+y|\vec{b}|^2 \vec{b}+z|\vec{c}|^2 \vec{c}-\vec{r}\left(|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2\right)+ \\
& |\vec{a}|^2 \vec{b}+|\vec{b}|^2 \vec{c}+|\vec{c}|^2 \vec{a}=0 \\
& \Rightarrow|\vec{a}|^2(x \vec{a}+y \vec{b}+z \vec{c})-3|\vec{a}|^2 \vec{r}+|\vec{a}|^2(\vec{a}+\vec{b}+\vec{c})=0 \\
& \Rightarrow 3 \vec{r}-\vec{r}=\vec{a}+\vec{b}+\vec{c} \\
& \Rightarrow \vec{r}=\frac{1}{2}(\vec{a}+\vec{b}+\vec{c})
\end{aligned}
$
Hence, the answer is $\frac{1}{2}(\vec{a}+\vec{b}+\vec{c})$

Example 5: Let three vector $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}, \vec{a} \cdot \vec{b}=7$ and $\vec{b}$ is perpendicular to $\vec{c}$, where $\vec{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{k}$. Then the value of $2|\vec{a}+\vec{b}+\vec{c}|_{\text {is }}^2$ $\qquad$
[JEE MAINS 2021]

Solution:

$
\begin{aligned}
\vec{c} & =\lambda(\vec{b} \times(\vec{a} \times \vec{b})) \\
& =\lambda((\vec{b} \cdot \vec{b}) \vec{b}-(\vec{b} \cdot \vec{a}) \vec{b}) \\
& =\lambda(5(-\hat{i}+\hat{j}+\hat{k})+2 \hat{i}+\hat{k}) \\
& =\lambda(-3 \hat{i}+5 \hat{j}+6 \hat{k}) \\
\vec{c} & \cdot \vec{a}=7 \Rightarrow 3 \lambda+5 \lambda+6 \lambda=7 \\
\Rightarrow & \lambda=\frac{1}{2} \\
\therefore & 2\left|\left(\frac{-3}{2}-1+2\right) \hat{i}+\left(\frac{5}{2}+1\right) \hat{j}+(3+1+1) \hat{k}\right|^2 \\
& =2\left(\frac{1}{4}+\frac{49}{4}+25\right)=25+50=75
\end{aligned}
$
Hence, the answer is 75

List of Topics Related to Vector Algebra

Provided below is the list of topics which are related to vector triple product in vector algebra, to boost your understanding and strengthen your concepts:

Types of Vectors

Vectors and Scalars

Addition of Vectors and Subtraction of Vectors

Multiplication Of Vectors by a Scalar Quantity

Components Of A Vector Along And Perpendicular To Another Vector

NCERT Resources

This section offers well-organized NCERT-based resources, including clear notes and step-by-step solutions, to help you study strictly according to the syllabus. It focuses on building strong conceptual clarity in Vector Algebra.

NCERT Maths Class 12th Notes for Chapter 10 - Vector Algebra

NCERT Maths Class 12th Solutions for Chapter 10 - Vector Algebra

NCERT Maths Class 12th Exemplar Solutions for Chapter 10 - Vector Algebra

Practice Questions based on Vector Triple Product

This section includes carefully selected practice questions based on the Vector Triple Product to help you apply formulas and identities with confidence. It is designed to strengthen your problem-solving skills and improve accuracy through regular practice.

Proof Of The Vector Triple Product- Practice Question MCQ

We have provided below the practice questions based on the topics related to Vector Triple Product: