Question

if a vector 2i^+3j^ + 8k^ is a perpendicular to d vector 4k^ - 4i^ + ak^. then d value of a is ???

Answer 1

you did little mistake in typing in question, question is ---> If a vector 2i + 3j + 8k is perpendicular to the vector 4 j- 4i + ak, then the value of a.

answer : a = -1/2

if two vectors are perpendicular to each other, then dot product of given vectors will be zero.

for instance, two vectors $A=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $B=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ are perpendicular to each other.

then, A.B = 0

or, $(a_1\hat{i}+a_2\hat{j}+a_3\hat{k}).(b_1\hat{i}+b_2\hat{j}+b_3\hat{k})=0$

or, $a_1a_2+b_1b_2+c_1c_2=0$

now come to question,

given, a vector $2\hat{i}+3\hat{j}+8\hat{k}$ is perpendicular to vector $4\hat{j}-4\hat{i}+a\hat{k}$ or, $-4\hat{i}+4\hat{j}+a\hat{k}$.

so, 2 × (-4) + 3 × 4 + 8 × a = 0

or, -8 + 12 + 8a = 0

or, 4 + 8a = 0

or, a = -1/2

Answer 2

given, a vector 2\hat{i}+3\hat{j}+8\hat{k} is perpendicular to vector 4\hat{j}-4\hat{i}+a\hat{k} or, -4\hat{i}+4\hat{j}+a\hat{k}.

so, 2 × (-4) + 3 × 4 + 8 × a = 0

or, -8 + 12 + 8a = 0

or, 4 + 8a = 0

or, a = -1/2