Question

which components are a possible representation of vector w if the magnitude of vector -3w is ||-3w|| = 15?
< 1, -9 >
< -3, 4 >
< 4, 3 >
< -5, -3 >
< 0, -5 >

Explanation:

Step1: Recall Magnitude Property

For a vector \(\mathbf{v}\) and scalar \(k\), \(||k\mathbf{v}|| = |k| \cdot ||\mathbf{v}||\). Given \(|| - 3\mathbf{w}|| = 15\), we have \(|-3| \cdot ||\mathbf{w}|| = 15\), so \(3||\mathbf{w}|| = 15\), hence \(||\mathbf{w}|| = 5\).

Step2: Magnitude Formula

The magnitude of a vector \(\mathbf{w} = \langle a, b
angle\) is \(||\mathbf{w}|| = \sqrt{a^2 + b^2}\). We need \(\sqrt{a^2 + b^2} = 5\) (i.e., \(a^2 + b^2 = 25\)).

Step3: Check Each Option

• Option \(\langle 1, -9

angle\): \(1^2 + (-9)^2 = 1 + 81 = 82
eq 25\)

• Option \(\langle -3, 4

angle\): \((-3)^2 + 4^2 = 9 + 16 = 25\) ✔️

• Option \(\langle 4, 5

angle\): \(4^2 + 5^2 = 16 + 25 = 41
eq 25\)

• Option \(\langle -5, -3

angle\): \((-5)^2 + (-3)^2 = 25 + 9 = 34
eq 25\)

• Option \(\langle 0, -5

angle\): \(0^2 + (-5)^2 = 0 + 25 = 25\) ✔️

Answer:

B. \(\langle -3, 4
angle\), E. \(\langle 0, -5
angle\)