Operations with Vectors: Learn It 2

Multiplying By a Scalar

While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

scalar multiplication

Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply [latex]\boldsymbol{v}[/latex] [latex]=\langle a,b\rangle[/latex] by [latex]k[/latex] , we have

[latex]k\boldsymbol{v}=\langle ka,kb\rangle[/latex]

Only the magnitude changes, unless [latex]k[/latex] is negative, and then the vector reverses direction.

Given vector [latex]\boldsymbol{v}[/latex] [latex]=\langle 3,1\rangle[/latex], find 3v, [latex]\frac{1}{2}[/latex] v, and −v.

Find the scalar multiple [latex]3\boldsymbol{u}[/latex] given [latex]\boldsymbol{u}[/latex] [latex]=\langle 5,4\rangle[/latex].

Given [latex]\boldsymbol{u}=\langle 3,-2\rangle[/latex] and [latex]\boldsymbol{v}=\langle -1,4\rangle[/latex], find a new vector w = 3u + 2v.

Evaluate [latex]2\boldsymbol{a} - 3\boldsymbol{b}[/latex] where [latex]\boldsymbol{a} = \boldsymbol{i} + 5\boldsymbol{j}[/latex] and [latex]\boldsymbol{b} = -2\boldsymbol{i} + 4\boldsymbol{j}[/latex].