Question

use the dot product to determine whether v and w are orthogonal. v = 8i + 8j, w = 8i - 8j select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. they are not orthogonal because the dot product is \boxed{}. b. they are orthogonal because the dot product is \boxed{}.

Explanation:

Step1: Recall dot product formula

For $\mathbf{v}=a_1\mathbf{i}+b_1\mathbf{j}$ and $\mathbf{w}=a_2\mathbf{i}+b_2\mathbf{j}$, $\mathbf{v}\cdot\mathbf{w}=a_1a_2 + b_1b_2$

Step2: Substitute values into formula

$\mathbf{v}\cdot\mathbf{w}=(8)(8)+(8)(-8)$

Step3: Calculate the dot product

$\mathbf{v}\cdot\mathbf{w}=64 - 64=0$

Step4: Check orthogonality rule

Vectors are orthogonal if their dot product is 0.

Answer:

B. They are orthogonal because the dot product is 0.