Question

-4v

= √((-4v₁)² + (-4v₂)²)

= √(12² + (-20)²)
= √(144 + 400)
= √544
≈ 23.32.

part d
compare the magnitudes of v and -4v. what relationship, if any, do you notice between their magnitudes?

Explanation:

To compare the magnitudes of \(\mathbf{v}\) and \(-4\mathbf{v}\), recall the property of vector magnitude: for a scalar \(c\) and vector \(\mathbf{v}\), \(\|c\mathbf{v}\| = |c| \cdot \|\mathbf{v}\|\). Here, \(c=-4\), so \(|-4| = 4\), meaning \(\|-4\mathbf{v}\| = 4\|\mathbf{v}\|\). This shows the magnitude of \(-4\mathbf{v}\) is 4 times the magnitude of \(\mathbf{v}\) (scaling a vector by a scalar multiplies its magnitude by the absolute value of the scalar).

Answer:

The magnitude of \(-4\mathbf{v}\) is 4 times the magnitude of \(\mathbf{v}\) (i.e., \(\|-4\mathbf{v}\| = 4\|\mathbf{v}\|\)).