Answer:
b = 6 or b = -6 (non-zero vectors)
b = 0 (zero vector)
Step-by-step explanation:
Two vectors [tex]\vec{a}=\langle a_1,a_2,a_3\rangle[/tex] and [tex]\vec{b}=\langle b_1,b_2,b_3\rangle[/tex] are orthogonal if their dot product is equal to 0, or in other words
[tex]a_1\cdot b_1+a_2\cdot b_2+a_3\cdot b_3=0[/tex]
In your case,
[tex]\vec{a}=\langle -46, b, 10\rangle\\ \\\vec{b}=\langle b,b^2,b\rangle[/tex]
Hence, if vectors a and b are orthogonal, then
[tex]-46\cdot b+b\cdot b^2+10\cdot b=0\\ \\-46b+b^3+10b=0\\ \\b^3-36b=0\\ \\b(b^2-36)=0\\ \\b(b-6)(b+6)=0\\ \\b=0\text{ or }b=6\text{ or }b=-6[/tex]
Note, then if b = 0, then [tex]\vec{b}=\langle 0,0,0\rangle[/tex] and zero-vector is orthogonal to any other vectors.
Thus, b = 6 or b = -6.
