Question

use the dot product to determine whether v and w are orthogonal. v = 2i - 6j, w = -8i + 12j select the correct choice below and, if necessary, fill in the answer box to complete your choice a. they are orthogonal because the dot product is \boxed{}. b. they are not 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

$\mathbf{v}\cdot\mathbf{w}=(2)(-8) + (-6)(12)$

Step3: Calculate the dot product

$\mathbf{v}\cdot\mathbf{w}=-16 - 72 = -88$

Step4: Check orthogonality rule

Vectors are orthogonal if their dot product is 0. Here, $\mathbf{v}\cdot\mathbf{w}=-88
eq0$.

Answer:

B. They are not orthogonal because the dot product is $-88$.