How to Find Scalar Multiplication of Vectors?

Scalar multiplication stretches a vector without turning it: multiply each component by the scalar. A positive scalar keeps the direction, a negative one reverses it, and the magnitude is multiplied by the absolute value of the scalar. Below: the properties and worked examples.

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.

Related Topics

• How to Find Vector Components
• How to Find Magnitude of Vectors
• Vectors Introduction

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:

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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. \(\color{blue}{-6v=-6i+42j+35k}\)