# How to Find Scalar Multiplication of Vectors?

Don't know how to scalar multiplication of vectors? The following step-by-step guide helps you learn how to find the scalar multiplication of vectors.

To multiply a vector by a scalar, multiply each component by a scalar. The product of a scalar with a vector is always a vector.

## A step-by-step guide to scalar multiplication of vectors

The scalar component of the vector is multiplied by the scalar component of each component of the vector.

If  $$u⃗=(u_1, u_2)$$ has a magnitude of $$|u⃗ |$$ and a direction of $$d$$, then:

$$\color{blue}{|nu⃗ |=n(u_1,u_2)=(nu_1,nu_2)}$$

where $$n$$ is a positive real integer, the magnitude is $$|nu⃗ |$$ and the direction is $$d$$.

If $$n$$ is negative, the direction of $$n$$u is the total opposite of $$d$$.

### Properties of Scalar Multiplication:

Allow u and v to be vectors, and $$c$$ and $$d$$ to be scalars. Then the properties listed below are true:

### Scalar Multiplication of Vectors – Example 1:

If u$$=(-2, 3)$$, find $$6$$u.

$$6$$u$$=6(-2,3)$$

$$=6(-2),6(3)$$

$$=(-12,18)$$

### Scalar Multiplication of Vectors – Example 2:

If a$$=3i+j+2k$$, find $$5$$a.

$$5$$a$$=5(3i+j+2k)$$

$$=5(3i)+5(j)+5(2k)$$

$$=15i+5j+10k$$

## Exercises for Scalar Multiplication of Vectors

1. If a$$=(-1,-8)$$, find $$4$$a.
2. If v$$=6i-5j+4k$$, find $$3$$v.
3. If a$$=7, -3$$, find $$-5$$a.
4. If v$$=i-7j-5k$$, find $$-6$$v.
1. $$\color{blue}{4a=(-4,-32)}$$
2. $$\color{blue}{3v=18i-15j+12k}$$
3. $$\color{blue}{-5a=(-35,15)}$$
4. $$\color{blue}{-6v=-6i+42j+35k}$$

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