What is the multiplication of a vector by a scalar?

It scales the vector’s magnitude by the scalar and possibly reverses its direction if the scalar is negative.

How do you multiply vectors?

Use scalar multiplication for resizing, dot product for projection, and cross product for a perpendicular vector.

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Multiplication of A Vector

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Multiplication of a Vector

In mathematics and physics, vector multiplication is a key operation used to analyze motion, force, and direction. It involves multiplying a vector either by a scalar (a number) or another vector. The result depends on the type of multiplication: scalar multiplication stretches or shrinks the vector, while vector multiplication yields a new vector or a scalar based on the rule used.

1.0Multiplication of Vectors Definition

Multiplication of a vector refers to scaling a vector by a number (scalar) or combining it with another vector using dot product or cross product. The operation you use determines whether the result is a scalar or another vector.

2.0Multiplication of a Vector by a Scalar (Number)

When you perform the multiplication of a vector by a scalar, you are changing the magnitude of the vector but not its direction (unless the scalar is negative, which reverses the direction).

Multiplication of a Vector by a Number Formula

If $A = a \hat{i} + b \hat{j}$ , and k is a scalar, then:

$k \cdot A = k (a \hat{i} + b \hat{j}) = (ka) \hat{i} + (kb) \hat{j}$

This is also called scalar multiplication of a vector.

3.0How to Multiply Vector Components

To multiply a vector by a number:

Multiply each component of the vector by the scalar.
Keep the same direction (or reverse it if the scalar is negative).
The result is another vector.

Example:
Given $A = 2 \hat{i} - 3 \hat{j}$ and scalar k = -2

$k \cdot A = - 2 (2 \hat{i} - 3 \hat{j}) = - 4 \hat{i} + 6 \hat{j}$

This is multiplication of vector components by a scalar.

4.0What Happens When a Vector Is Multiplied by a Scalar?

The magnitude of the vector changes.
The direction remains the same if scalar is positive.
The direction reverses if scalar is negative.
The vector becomes a zero vector if multiplied by 0.

5.0Multiplication of Vectors by Another Vector

There are two types of vector-by-vector multiplication:

1. Dot Product (Scalar Product)

Result: Scalar
Formula:

$A \cdot B = ∣ A ∣∣ B ∣ cos (θ)$

Application: Work done, projection

2. Cross Product (Vector Product)

Result: Vector
Formula:

$A \times B = ∣ A ∣∣ B ∣ sin (θ) \overset{n}{^}$

where It is a unit vector perpendicular to both $A$ and $B$ .

Application: Torque, rotational motion

6.0What Is the Rule for Vector Multiplication?

Type of Multiplication	Rule	Result
Scalar × Vector	Multiply scalar to each vector component	Vector
Vector · Vector (Dot)	Multiply magnitudes and cosine of angle	Scalar
Vector × Vector (Cross)	Multiply magnitudes and sine of angle, direction via right-hand rule	Vector

7.0Multiplication of Vectors Example

Example 1: Scalar Multiplication of $A = 3 \hat{i} + 4 \hat{j}, k = 2$

Solution:

$A = 2 (3 \hat{i} + 4 \hat{j}) = 6 \hat{i} + 8 \hat{j}$

Example 2: Dot Product of $A = ⟨ 2, 3 ⟩, B = ⟨ 4, - 1 ⟩$

Solution:

$A \cdot B = (2) (4) + (3) (- 1) = 8 - 3 = 5$

Example 3: Cross Product of $A = \hat{i} + 2 \hat{j}, B = 3 \hat{i} + \hat{j}$

Solution:

$A \times B = \hat{i} 13 \hat{j} 21 \hat{k} 00 = (0) \hat{i} - (0) \hat{j} + (1 \cdot 1 - 2 \cdot 3) \hat{k} = - 5 \hat{k}$

8.0JEE Advanced-Level Questions & Solutions on Multiplication of Vectors

Q1. Let $A = \hat{i} + 2 \hat{j} + \hat{k} B = 2 \hat{i} - \hat{j} + \hat{k}$ , and $C = a \hat{i} + b \hat{j} + c \hat{k}$ . If the vectors $A, B, C$ are coplanar, find the relation between a, b, c.

Solution:

Use the scalar triple product condition:

$A \cdot (B \times C) = 0$

Evaluate:

$B \otimes C \otimes = \hat{i} 2 a \hat{j} - 1 b \hat{k} 1 c B \times C = ((- 1) (c) - (1) (b)) \hat{i} - ((2) (c) - (1) (a)) \hat{j} + ((2) (b) - (- 1) (a)) \hat{k} B \times C = (- c - b) \hat{i} - (2 c - a) \hat{j} + (2 b + a) \hat{k}$

Now dot with $A$ :

=−5c + a + b(1)(-c - b) + 2(-2c + a) + 1(2b + a)

= - c - b - 4c + 2a + 2b + a = -5c + a + b

Set equal to 0:

a + b - 5c = 0

Answer: ca + b = 5c

Q2. Let A and B be $A = 3 \hat{i} + 4 \hat{j}, B = 5 \hat{i}$ . Find the projection of $A o n B$ .

Solution:

$Projection of A on B = \frac{A \cdot B}{∣ B B ∣}$

$Projection = \frac{A \cdot B}{∣ B ∣} = \frac{3 ( 5 )}{5} = 3$

Answer: 3

Q3. Find angle between two vectors A and B. $A = 2 \hat{i} + \hat{j} + 2 \hat{k}, B = \hat{i} - \hat{j} + \hat{k}$

Solution:

$A \cdot B = 2 (1) + 1 (- 1) + 2 (1) = 2 - 1 + 2 = 3$

$∣ A ∣ = 4 + 1 + 4 = 9 = 3, ∣ B ∣ = 1 + 1 + 1 = 3$

Using formula:

$cos θ = \frac{A \cdot B}{∣ A ∣∣ B ∣}$

$cos θ = \frac{3}{3 3} = \frac{1}{3} \Rightarrow θ = cos^{- 1} (\frac{1}{3})$

Answer: $θ = cos^{- 1} (\frac{1}{3})$

Q4. Find area of parallelogram using cross product of $u = 3 \hat{i} - \hat{j} + \hat{k}, v = \hat{i} + 2 \hat{j} - 2 \hat{k} .$

Solution:

$Area = ∣ u \times v ∣$

$u \times v = \hat{i} 31 \hat{j} - 1 2 \hat{k} 1 - 2 \Rightarrow u \times v = (- 1) (- 2) - (1) (2) \hat{i} - [3 (- 2) - (1) (1)] \hat{j} + [3 (2) - (- 1) (1)] \hat{k} \Rightarrow u \times v = (2 - 2) \hat{i} + (6 - 1) \hat{j} + (6 + 1) \hat{k} \Rightarrow ∣ u \times v ∣ = 0 \hat{i} + 5 \hat{j} + 7 \hat{k} \Rightarrow ∣ u \times v ∣ = 0^{2} + 25 + 49 = 74$

Area: $74$

Q5. Find shortest distance between:

$L_{1} : r = \hat{i} + 2 \hat{j} + \hat{k} + λ (\hat{i} + \hat{j}), L_{2} : r = 2 \hat{i} + \hat{j} + 2 \hat{k} + μ (2 \hat{i} - \hat{j} + \hat{k})$

Solution:

Let:

$a_{1} = \hat{i} + 2 \hat{j} + \hat{k}$
$a_{2} = 2 \hat{i} + \hat{j} + 2 \hat{k}$
$b_{1} = \hat{i} + \hat{j}, b_{2} = 2 \hat{i} - \hat{j} + \hat{k}$

Solution:

$a_{2} - a_{1} = \hat{i} - \hat{j} + \hat{k} b_{1} \times b_{2} = \hat{i} 12 \hat{j} 1 - 1 \hat{k} 01 b_{1} \times b_{2} = (1) (1) - (0) (- 1) \hat{i} - (1) (1) - (0) (2) \hat{j} + (1) (- 1) - (1) (2) \hat{k} b_{1} \times b_{2} = \hat{i} - \hat{j} - 3 \hat{k}$

Using formula of shortest distance between two lines:

$d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}$

$d = \frac{∣ ( i ^ - j ^ + k ^ ) \cdot ( i ^ - j ^ - 3 k ^ ) ∣}{∣ i ^ - j ^ - 3 k ^ ∣} d = \frac{∣1 + 1 - 3∣}{1 + 1 + 9} d = \frac{∣ - 1∣}{11} d = \frac{1}{11}$

Answer: $\frac{1}{11}$

Q6. Find a unit vector perpendicular to both $A = 2 \hat{i} + 3 \hat{j} - \hat{k}, B = \hat{i} - \hat{j} + 4 \hat{k}$

Solution:

Unit vector = $\frac{A \times B}{∣ A \times B ∣}$

$A B = \hat{i} 21 \hat{j} 3 - 1 \hat{k} - 1 4 A \times B = ((3) (4) - (- 1) (- 1)) \hat{i} - ((2) (4) - (- 1) (1)) \hat{j} + ((2) (- 1) - (3) (1)) \hat{k} A \times B = (12 - 1) \hat{i} - (8 + 1) \hat{j} + (- 2 - 3) \hat{k} A \times B = 11 \hat{i} - 9 \hat{j} - 5 \hat{k}$

$\Rightarrow Unit Vector = \frac{11 i ^ - 9 j ^ - 5 k ^}{121 + 81 + 25} = \frac{11 i ^ - 9 j ^ - 5 k ^}{227}$

Answer: $\frac{11 i ^ - 9 j ^ - 5 k ^}{227}$

Q7. If $A = \hat{i} + 2 \hat{j}, B = 2 \hat{i} - 3 \hat{j} , f in d a v ec t or C s u c h t ha t : A \cdot C = 0, B \cdot C = 0$

Solution:

Let $C = x \hat{i} + y \hat{j} + z \hat{k}$

Then,

$A \cdot C B \cdot C = x + 2 y = 0 \Rightarrow x = - 2 y B \cdot C = 2 x - 3 y = 0 \Rightarrow 2 (- 2 y) - 3 y = - 4 y - 3 y = - 7 y = 0 \Rightarrow y = 0 \Rightarrow x = 0$

So

$C = z \hat{k}$

Answer: Any vector along .

Q8: What is the multiplication of vectors formula?

$S c a l a r : k \cdot A = k A_{x} \hat{i} + k A_{y} \hat{j}$
$Do t : A \cdot B = ∣ A ∣∣ B ∣ cos θ$
$C ross : A \times B = ∣ A ∣∣ B ∣ sin θ \overset{n}{^}$

9.0Practice Questions on Vector Multiplication

Multiply $A = (2, - 1) b ysc a l a r - 3$ .
Find the dot product of $A = (4, 2) an d B = (- 1, 5)$ .
Calculate $A \times B w h ere A = (0, 2, 1), B = (3, 1, 4)$ .
If $V = 5 \hat{i} - \hat{j}, f in d 0 \cdot V$ .
What happens to a vector when multiplied by -1?

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Multiplication of a Vector

In mathematics and physics, vector multiplication is a key operation used to analyze motion, force, and direction. It involves multiplying a vector either by a scalar (a number) or another vector. The result depends on the type of multiplication: scalar multiplication stretches or shrinks the vector, while vector multiplication yields a new vector or a scalar based on the rule used.

1.0Multiplication of Vectors Definition

Multiplication of a vector refers to scaling a vector by a number (scalar) or combining it with another vector using dot product or cross product. The operation you use determines whether the result is a scalar or another vector.

2.0Multiplication of a Vector by a Scalar (Number)

When you perform the multiplication of a vector by a scalar, you are changing the magnitude of the vector but not its direction (unless the scalar is negative, which reverses the direction).

Multiplication of a Vector by a Number Formula

If $A = a \hat{i} + b \hat{j}$ , and k is a scalar, then:

$k \cdot A = k (a \hat{i} + b \hat{j}) = (ka) \hat{i} + (kb) \hat{j}$

This is also called scalar multiplication of a vector.

3.0How to Multiply Vector Components

To multiply a vector by a number:

Multiply each component of the vector by the scalar.
Keep the same direction (or reverse it if the scalar is negative).
The result is another vector.

Example:
Given $A = 2 \hat{i} - 3 \hat{j}$ and scalar k = -2

$k \cdot A = - 2 (2 \hat{i} - 3 \hat{j}) = - 4 \hat{i} + 6 \hat{j}$

This is multiplication of vector components by a scalar.

4.0What Happens When a Vector Is Multiplied by a Scalar?

The magnitude of the vector changes.
The direction remains the same if scalar is positive.
The direction reverses if scalar is negative.
The vector becomes a zero vector if multiplied by 0.

5.0Multiplication of Vectors by Another Vector

There are two types of vector-by-vector multiplication:

1. Dot Product (Scalar Product)

Result: Scalar
Formula:

$A \cdot B = ∣ A ∣∣ B ∣ cos (θ)$

Application: Work done, projection

2. Cross Product (Vector Product)

Result: Vector
Formula:

$A \times B = ∣ A ∣∣ B ∣ sin (θ) \overset{n}{^}$

where It is a unit vector perpendicular to both $A$ and $B$ .

Application: Torque, rotational motion

6.0What Is the Rule for Vector Multiplication?

Type of Multiplication	Rule	Result
Scalar × Vector	Multiply scalar to each vector component	Vector
Vector · Vector (Dot)	Multiply magnitudes and cosine of angle	Scalar
Vector × Vector (Cross)	Multiply magnitudes and sine of angle, direction via right-hand rule	Vector

7.0Multiplication of Vectors Example

Example 1: Scalar Multiplication of $A = 3 \hat{i} + 4 \hat{j}, k = 2$

Solution:

$A = 2 (3 \hat{i} + 4 \hat{j}) = 6 \hat{i} + 8 \hat{j}$

Example 2: Dot Product of $A = ⟨ 2, 3 ⟩, B = ⟨ 4, - 1 ⟩$

Solution:

$A \cdot B = (2) (4) + (3) (- 1) = 8 - 3 = 5$

Example 3: Cross Product of $A = \hat{i} + 2 \hat{j}, B = 3 \hat{i} + \hat{j}$

Solution:

$A \times B = \hat{i} 13 \hat{j} 21 \hat{k} 00 = (0) \hat{i} - (0) \hat{j} + (1 \cdot 1 - 2 \cdot 3) \hat{k} = - 5 \hat{k}$

8.0JEE Advanced-Level Questions & Solutions on Multiplication of Vectors

Q1. Let $A = \hat{i} + 2 \hat{j} + \hat{k} B = 2 \hat{i} - \hat{j} + \hat{k}$ , and $C = a \hat{i} + b \hat{j} + c \hat{k}$ . If the vectors $A, B, C$ are coplanar, find the relation between a, b, c.

Solution:

Use the scalar triple product condition:

$A \cdot (B \times C) = 0$

Evaluate:

$B \otimes C \otimes = \hat{i} 2 a \hat{j} - 1 b \hat{k} 1 c B \times C = ((- 1) (c) - (1) (b)) \hat{i} - ((2) (c) - (1) (a)) \hat{j} + ((2) (b) - (- 1) (a)) \hat{k} B \times C = (- c - b) \hat{i} - (2 c - a) \hat{j} + (2 b + a) \hat{k}$

Now dot with $A$ :

=−5c + a + b(1)(-c - b) + 2(-2c + a) + 1(2b + a)

= - c - b - 4c + 2a + 2b + a = -5c + a + b

Set equal to 0:

a + b - 5c = 0

Answer: ca + b = 5c

Q2. Let A and B be $A = 3 \hat{i} + 4 \hat{j}, B = 5 \hat{i}$ . Find the projection of $A o n B$ .

Solution:

$Projection of A on B = \frac{A \cdot B}{∣ B B ∣}$

$Projection = \frac{A \cdot B}{∣ B ∣} = \frac{3 ( 5 )}{5} = 3$

Answer: 3

Q3. Find angle between two vectors A and B. $A = 2 \hat{i} + \hat{j} + 2 \hat{k}, B = \hat{i} - \hat{j} + \hat{k}$

Solution:

$A \cdot B = 2 (1) + 1 (- 1) + 2 (1) = 2 - 1 + 2 = 3$

$∣ A ∣ = 4 + 1 + 4 = 9 = 3, ∣ B ∣ = 1 + 1 + 1 = 3$

Using formula:

$cos θ = \frac{A \cdot B}{∣ A ∣∣ B ∣}$

$cos θ = \frac{3}{3 3} = \frac{1}{3} \Rightarrow θ = cos^{- 1} (\frac{1}{3})$

Answer: $θ = cos^{- 1} (\frac{1}{3})$

Q4. Find area of parallelogram using cross product of $u = 3 \hat{i} - \hat{j} + \hat{k}, v = \hat{i} + 2 \hat{j} - 2 \hat{k} .$

Solution:

$Area = ∣ u \times v ∣$

$u \times v = \hat{i} 31 \hat{j} - 1 2 \hat{k} 1 - 2 \Rightarrow u \times v = (- 1) (- 2) - (1) (2) \hat{i} - [3 (- 2) - (1) (1)] \hat{j} + [3 (2) - (- 1) (1)] \hat{k} \Rightarrow u \times v = (2 - 2) \hat{i} + (6 - 1) \hat{j} + (6 + 1) \hat{k} \Rightarrow ∣ u \times v ∣ = 0 \hat{i} + 5 \hat{j} + 7 \hat{k} \Rightarrow ∣ u \times v ∣ = 0^{2} + 25 + 49 = 74$

Area: $74$

Q5. Find shortest distance between:

$L_{1} : r = \hat{i} + 2 \hat{j} + \hat{k} + λ (\hat{i} + \hat{j}), L_{2} : r = 2 \hat{i} + \hat{j} + 2 \hat{k} + μ (2 \hat{i} - \hat{j} + \hat{k})$

Solution:

Let:

$a_{1} = \hat{i} + 2 \hat{j} + \hat{k}$
$a_{2} = 2 \hat{i} + \hat{j} + 2 \hat{k}$
$b_{1} = \hat{i} + \hat{j}, b_{2} = 2 \hat{i} - \hat{j} + \hat{k}$

Solution:

$a_{2} - a_{1} = \hat{i} - \hat{j} + \hat{k} b_{1} \times b_{2} = \hat{i} 12 \hat{j} 1 - 1 \hat{k} 01 b_{1} \times b_{2} = (1) (1) - (0) (- 1) \hat{i} - (1) (1) - (0) (2) \hat{j} + (1) (- 1) - (1) (2) \hat{k} b_{1} \times b_{2} = \hat{i} - \hat{j} - 3 \hat{k}$

Using formula of shortest distance between two lines:

$d = \frac{∣ ( a _{2} - a _{1} ) \cdot ( b _{1} \times b _{2} ) ∣}{∣ b _{1} \times b _{2} ∣}$

$d = \frac{∣ ( i ^ - j ^ + k ^ ) \cdot ( i ^ - j ^ - 3 k ^ ) ∣}{∣ i ^ - j ^ - 3 k ^ ∣} d = \frac{∣1 + 1 - 3∣}{1 + 1 + 9} d = \frac{∣ - 1∣}{11} d = \frac{1}{11}$

Answer: $\frac{1}{11}$

Q6. Find a unit vector perpendicular to both $A = 2 \hat{i} + 3 \hat{j} - \hat{k}, B = \hat{i} - \hat{j} + 4 \hat{k}$

Solution:

Unit vector = $\frac{A \times B}{∣ A \times B ∣}$

$A B = \hat{i} 21 \hat{j} 3 - 1 \hat{k} - 1 4 A \times B = ((3) (4) - (- 1) (- 1)) \hat{i} - ((2) (4) - (- 1) (1)) \hat{j} + ((2) (- 1) - (3) (1)) \hat{k} A \times B = (12 - 1) \hat{i} - (8 + 1) \hat{j} + (- 2 - 3) \hat{k} A \times B = 11 \hat{i} - 9 \hat{j} - 5 \hat{k}$

$\Rightarrow Unit Vector = \frac{11 i ^ - 9 j ^ - 5 k ^}{121 + 81 + 25} = \frac{11 i ^ - 9 j ^ - 5 k ^}{227}$

Answer: $\frac{11 i ^ - 9 j ^ - 5 k ^}{227}$

Q7. If $A = \hat{i} + 2 \hat{j}, B = 2 \hat{i} - 3 \hat{j} , f in d a v ec t or C s u c h t ha t : A \cdot C = 0, B \cdot C = 0$

Solution:

Let $C = x \hat{i} + y \hat{j} + z \hat{k}$

Then,

$A \cdot C B \cdot C = x + 2 y = 0 \Rightarrow x = - 2 y B \cdot C = 2 x - 3 y = 0 \Rightarrow 2 (- 2 y) - 3 y = - 4 y - 3 y = - 7 y = 0 \Rightarrow y = 0 \Rightarrow x = 0$

So

$C = z \hat{k}$

Answer: Any vector along .

Q8: What is the multiplication of vectors formula?

$S c a l a r : k \cdot A = k A_{x} \hat{i} + k A_{y} \hat{j}$
$Do t : A \cdot B = ∣ A ∣∣ B ∣ cos θ$
$C ross : A \times B = ∣ A ∣∣ B ∣ sin θ \overset{n}{^}$

9.0Practice Questions on Vector Multiplication

Multiply $A = (2, - 1) b ysc a l a r - 3$ .
Find the dot product of $A = (4, 2) an d B = (- 1, 5)$ .
Calculate $A \times B w h ere A = (0, 2, 1), B = (3, 1, 4)$ .
If $V = 5 \hat{i} - \hat{j}, f in d 0 \cdot V$ .
What happens to a vector when multiplied by -1?

Frequently Asked Questions

What is the multiplication of a vector by a scalar?

How do you multiply vectors?

What happens when a vector is multiplied by zero?

Frequently Asked Questions

What is the multiplication of a vector by a scalar?

How do you multiply vectors?

What happens when a vector is multiplied by zero?

Join ALLEN!

Multiplication of a Vector

1.0Multiplication of Vectors Definition

2.0Multiplication of a Vector by a Scalar (Number)

3.0How to Multiply Vector Components

4.0What Happens When a Vector Is Multiplied by a Scalar?

5.0Multiplication of Vectors by Another Vector

6.0What Is the Rule for Vector Multiplication?

7.0Multiplication of Vectors Example

8.0JEE Advanced-Level Questions & Solutions on Multiplication of Vectors

9.0Practice Questions on Vector Multiplication

Table of Contents

Join ALLEN!

Multiplication of a Vector

1.0Multiplication of Vectors Definition

2.0Multiplication of a Vector by a Scalar (Number)

3.0How to Multiply Vector Components

4.0What Happens When a Vector Is Multiplied by a Scalar?

5.0Multiplication of Vectors by Another Vector

6.0What Is the Rule for Vector Multiplication?

7.0Multiplication of Vectors Example

8.0JEE Advanced-Level Questions & Solutions on Multiplication of Vectors

9.0Practice Questions on Vector Multiplication

Table of Contents