How to Find 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.

Related Topics

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

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