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Lyrx [107]
3 years ago
7

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 =
Mathematics
1 answer:
bearhunter [10]3 years ago
8 0

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.

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