# 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.

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## 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

- If
**a**\(=(-1,-8)\), find \(4\)**a**. - If
**v**\(=6i-5j+4k\), find \(3\)**v**. - If
**a**\(=7, -3\), find \(-5\)**a**. - If
**v**\(=i-7j-5k\), find \(-6\)**v**.

- \(\color{blue}{4a=(-4,-32)}\)
- \(\color{blue}{3v=18i-15j+12k}\)
- \(\color{blue}{-5a=(-35,15)}\)
- \(\color{blue}{-6v=-6i+42j+35k}\)

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