Question

For what values of b are the vectors [−18, b, 9] and [b, b2, b] orthogonal? (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) b =

Answer 1

Answer:

Therefore the given vectors are orthogonal for b = 0,±3.

Step-by-step explanation:

If  $\vec a$ and  $\vec b$ are two vectors orthogonal, then the dot product of $\vec a$ and $\vec b$ will be zero.

i.e $\vec a. \vec b =0$

If  $\vec a = x_1\hat i+y_1\hat j +z_1\hat k$  and $\vec b = x_2\hat i+y_2\hat j +z_2\hat k$

$\vec a. \vec b=( x_1\hat i+y_1\hat j +z_1\hat k).( x_2\hat i+y_2\hat j +z_2\hat k)$

     $=x_1 x_2+y_1y_2+z_1z_2$

Given two vectors are (-18,b,9) and (b,b²,b)

Let

$\vec P= -18 \hat i+b\hat j +9 \hat k$

and

$\vec Q = b \hat i+b^2 \hat j +b\hat k$

Therefore,

$\vec P.\vec Q$

$=( -18 \hat i+b\hat j +9 \hat k).( b \hat i+b^2 \hat j +b\hat k)$

=(-18).b+b.b²+9.b

= -18b+b³+9b

= b³-9b

Since $\vec P$ and $\vec Q$ are orthogonal. Then $\vec P.\vec Q$ = 0.

Therefore,

b³-9b= 0

⇒b(b²-9)=0

⇒b =0 or b²=9

⇒b=0 or b =±3

Therefore the given vectors are orthogonal for b = 0,±3.