Question

suppose u · v = 0. which vectors could represent u and v?
○ u = (3, 4) and v = (4, -3)
○ u = (3, 4) and v = (-4, -3)
○ u = (-3, 4) and v = (4, -3)
○ u = (-3, -4) and v = (-4, -3)

Explanation:

Step1: Recall the dot - product formula

If \( \mathbf{u}=\langle a,b
angle \) and \( \mathbf{v}=\langle c,d
angle \), then \( \mathbf{u}\cdot\mathbf{v}=ac + bd \)

Step2: Calculate the dot - product for each option

• Option A:

For \( \mathbf{u}=\langle 3,4
angle \) and \( \mathbf{v}=\langle 4,-3
angle \), \( \mathbf{u}\cdot\mathbf{v}=(3\times4)+(4\times(- 3))\)
\(=12-12 = 0\)

• Option B:

For \( \mathbf{u}=\langle 3,4
angle \) and \( \mathbf{v}=\langle - 4,-3
angle \), \( \mathbf{u}\cdot\mathbf{v}=(3\times(-4))+(4\times(-3))\)
\(=-12 - 12=-24
eq0\)

• Option C:

For \( \mathbf{u}=\langle - 3,4
angle \) and \( \mathbf{v}=\langle 4,-3
angle \), \( \mathbf{u}\cdot\mathbf{v}=(-3\times4)+(4\times(-3))\)
\(=-12-12=-24
eq0\)

• Option D:

For \( \mathbf{u}=\langle - 3,-4
angle \) and \( \mathbf{v}=\langle - 4,-3
angle \), \( \mathbf{u}\cdot\mathbf{v}=(-3\times(-4))+(-4\times(-3))\)
\(=12 + 12=24
eq0\)

Answer:

A. \( \mathbf{u} = \langle 3,4
angle \) and \( \mathbf{v}=\langle 4, - 3
angle \)