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.

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.

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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.
This image has an empty alt attribute; its file name is answers.png
  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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