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**Multiplication by a Scalar -Explanation and Examples**

**Multiplication by a scalar **is a way of changing the magnitude or direction of a vector. Put, it is

Recall that a scalar is just a real number. Multiplying a vector by a scalar causes a change in the scale of that vector.

In this topic, we will discuss the following aspects of scalar multiplication:

- What is Scalar Multiplication?
- How to Multiply a Vector by a Scalar?
- Multiplying a Vector by a Scalar

## What is Scalar Multiplication?

Scalar multiplication involves multiplying a given quantity by a scalar quantity. If the given quantity is scalar, the multiplication yields another scalar quantity. But, if the quantity is a vector, multiplication with a scalar gives a vector output.

**For example**, the multiplication of a scalar C with a vector **A** will yield another vector. We write this operation as:

C***A = **C**A**

In the above example, the resultant vector C**A** is the scaled version of vector **A** whose magnitude is C times the magnitude of the original vector **A**. Its direction is determined by the value of C in the following way:

- If C > 0, then the resultant vector C
**A**will have the same direction as the vector**A.** - If C <0, then the resultant vector is:

-C***A = –**C**A**The negative sign will reverse the direction of the resultant vector relative to the reference vector**A.** - If C = 0, then the multiplication yields a zero vector as:

0***A = 0**

Note that if C = 1, then multiplying any vector by C keeps that vector unchanged.

1***A **=** A**

## How to Multiply a Vector by a Scalar?

Suppose a vector **P** is expressed as the column vector:

**P** = (x1, y1).

Multiplying it by a scalar means scaling each component of the vector **P** by C as follows:

C***P **= C (x1, y1)

C***P **= (Cx1, Cy1)

Now, the magnitude of the resultant vector can be found in the same way that we can find the magnitude of the vector **P:**

|C***P**| = √(Cx1)^2 + (CX2)^2

## Multiplying a Vector by a Scalar

In this section, we will discuss some important properties of scalar multiplication. Note that these properties are true whether a scalar is multiplied by a vector or by another scalar.

Let’s first consider two vectors, **A** and **B,** and two scalars, c, and d. Then the following properties hold:

- |c
**A**| = |c|*|**A|.**The magnitude of the resultant scaled vector is equal to the absolute value of the scalar times the magnitude. - Associative property: c(d
**B**) = (cd)***B** - Commutative property: c*
**A**=**A***c - Distributive property: (c + d)
**A =**c***A +**d***A**

d***** (**A **+ **B**) = d***A **+ d*** B**

### Examples

In this section, we will discuss some examples and their step-by-step solutions in order to help establish a better understanding of scalar multiplication.

*Example 1 *

A car is moving with a velocity of **V **= 30 m/s towards the North. Determines the vector that is twice this vector.

__Solution__

From the given data, we have the following information:

**V** = 30 m/s North.

To determine the vector equal to twice this vector, we multiply the given vector by the scalar value 2. This gives us:

2* **V** = 2 * (30 m/s)

2**V** = 60 m/s, North

Since the given scalar value is positive, the direction of **V **is not affected**. **It does, however, change its magnitude to two times the initial value. Thus, the car will keep moving North with twice its initial velocity.

*Example 2*

Given a vector **S** = (2, 3), determine and sketch 2***S. **What are the magnitude and the direction of the vector 2**S**?

__Solution__

The given vector **S** is a column vector, and the scalar quantity is 2. Multiplying the vector S by 2 gives us:

2***S** = 2* (2, 3)

Multiplying each of the components of the vector **S** by 2 gives us:

2***S** = (2*2, 2* 3)

2***S** = (4, 6).

Next, we determine and compare the magnitudes of both the vectors:

|**S**| = √2^2 + 3^2

|**S**| = √4 + 9

|**S**| = √13

The magnitude of vector 2**S **is :

|2**S**| = √4^2 + 6^2

|2**S**| = √16 + 36

|2**S**| = √52

|2**S**| = √4*13

|2**S**| = 2*(√13)

It can be clearly observed from the last equation that the scalar multiplication has resulted doubled the magnitude of the vector **S.**

The image given below shows the two vectors, **S** and 2**S**. It can be seen that the direction of the vector 2**S** is parallel to that of the vector **S**. This further verifies that scaling a vector by a positive quantity only changes the magnitude and does not change the direction.

*Example 3*

Given a vector **S** = (2, 3), determine and sketch -2***S. **Find the magnitude and direction of the vector -2**S**.

__Solution__

The given vector **S** is a column vector, and the scalar quantity is 2. Multiplying the vector S by 2 gives us:

-2***S** = -2* (2, 3)

Multiplying each of the components of the vector **S** by 2 gives us:

-2***S** = (-2*2, -2* 3)

-2***S** = (-4, -6).

Next, we determine and compare the magnitudes of both the vectors:

|**S**| = √2^2 + 3^2

|**S**| = √4 + 9

|**S**| = √13

The magnitude of vector -2**S **is :

|-2**S**| = √(-4)^2 + (-6)^2

|-2**S**| = √16 + 36

|-2**S**| = √52

|-2**S**| = √4*13

|-2**S**| = 2*(√13)

It can be clearly observed from the last equation that the scalar multiplication has doubled the magnitude of the vector **S**. Also, the negative sign has no impact on the magnitude of the vector -2**S.**

The image given below shows the two vectors **S** and -2**S. **It can be seen that the direction of the vector -2**S** is opposite that of the vector **S**. This further verifies that scaling a vector by a negative quantity does not affect its magnitude (i.e., vectors 2**S** and -2**S** have the same magnitude) but does reverse the direction.

*Example 4*

Given a vector **A** = (-4, 6), determine and sketch the vector 1/2***A**.

__Solution__

The given vector **A** is a column vector, and the scalar quantity is 1/2. Multiplying the vector **A** by 1/2 gives us:

1/2***A** = 1/2* (-4, 6).

Simplifying gives us:

1/2***A** = (1/2*(-4),1/2*(6))

1/2***A** = (-2, 3).

Next, we determine and compare the magnitudes of both vectors:

|**A**| = √-4^2 + 6^2

|**A**| = √16 + 36

|**A**| = √52

|**A**| = 2*(√13)

The magnitude of vector 1/2**A **is :

|1/2**A**| = √-2^2 + 3^2

|1/2**A**| = √4 + 9

|1/2**A**| = √13

Multiplication by a scalar with a value of one half thus decreased the magnitude of the original vector by one half.

The image given below shows the two vectors **A** and ½** A. **Both vectors have the same direction but different magnitudes.

*Example 5*

Given a vector **m **= 5i + 6j +3 in the orthogonal system, determine the resultant vector if **m** is multiplied by 7.

__Solution__

In this scenario, the resultant vector can be obtained by simply multiplying the given vector by 7:

7**m** = 7 *(5i + 6j +3)

7**m** = (7*5i + 7*6j + 7*3)

7**m** = 35i + 42j + 21

The resultant vector has a 7 times greater magnitude than the original vector **m **but no change in direction.

*Practice Questions*

- Given a vector
**M**= 10 m East, determine the resultant vector obtained by multiplying the given vector by 3. - Given a vector
**N**= 15 m North, determine the resultant vector obtained by multiplying the given vector by -4. - Let
**u**= (-1, 4). Find 5**u**. - Let
**v**= (3, 9). Find -1/3**v**. - Given a vector
**b**= -3i + 2j +2 in the orthogonal system, find 5**b**.

__Answers__

- 3
**M**= 30 m, East. - -4
**N**= -60 m, South. - 5
**u**= (-5, 20), |**u**| = √17, |5**u**| = 5*√17. The direction of**u**and 5**u**is the same. - -1/3
**v**= (-1, -3), |**v**| = 3*√10, |-1/3**v**| = √10, the direction of the vector -1/3**v**is opposite to the direction of the vector**v**. - 5
**b**= -15i + 10j + 10