Vectors: Find the possible values of the constant

Question

For this question do I have to assume a.b=0?

how can I find the value of constants?

my understanding is: <,8,3+1>.<+1,-1,-2>=0

how do I proceed from here to find the values of constant ?

Answer 1

You have started correctly. The equation is $\alpha (\alpha +1)+8(\alpha -1)-2(3\alpha+1)=0$. This c an be written as $\alpha ^{2}+3\alpha-10=0$. Solve this quadratic equation to find the two values of $\alpha$. That is $(\alpha-2)(\alpha+5)=0$ and hence $\alpha=2, $ $\alpha=-5.$

Answer 2

Yes if the vectors are perpendicular, their dot product is $0$. Then use a$\cdot$b $=\sum_{i=1}^{n}a_i b_i$. Then $\alpha(\alpha+1)+8(\alpha-1)-2(3\alpha+1)=0$. Can you proceed?