Answer :
Answer:
The values of b are [-2,0,2] for which the vectors are orthogonal.
Step-by-step explanation:
Given:
Vector [-9,b,5] and Vector [b,b^2,b].
According to the question:
Both the above vectors are orthogonal we have to find the value of b.
Orthogonal vectors:
- Are the vectors which, are at right angles to each other.
- Meaning they are perpendicular to each other.
- And their,Dot product is equivalent to zero.
So we have to calculate by dot product and make it equal to zero and then have to solve the b values.
Dot Product:
For vectors a and b where [tex]a=(a_1,a_2,a_3)[/tex] and [tex]b=(b_1,b_2,b_3)[/tex] the dot product is [tex](a.b)=(a_1b_1+a_2b_2+a_3b_3)[/tex].
Solving.
⇒ [tex](-9,b,5).(b,b^2,b)[/tex]
⇒ [tex](-9b+b^3+5b) =0[/tex]
⇒ [tex](b^3-4b)=0[/tex]
⇒ [tex]b(b^2-4)=0[/tex] ...taking b as common.
⇒ [tex]b(b^2-2^2)=0[/tex]
⇒ [tex]b(b-2)(b+2)=0[/tex] ...using algebraic identity, [tex]a^2-b^2=(a-b)(a+b)[/tex].
⇒ Equating with zero individually.
⇒ [tex]b=-2,0,2[/tex]
So the values of b are [-2,0,2] for which the given vectors are orthogonal.
The value of b for the vectors to be orthogonal is (0, -2, 2)
Orthogonal coordinates
For the given coordinates to be orthogonal, the dot product of its coordinate must be equal to zero as shown:
Given the coordinate vectors [−9, b, 5] and [b, b2, b]
Taking the product of the coordinates
-9b + b³ + 5b = 0
b³ - 4b =0
b(b²-4) = 0
b = 0 and b² - 4 = 0
b = 0, -2 and 2
Hence the value of b for the vectors to be orthogonal is (0, -2, 2)
Learn more on dot product here: https://brainly.com/question/9956772