Question

Find the value of "X" in a way that A= 12î + xj + 6k and B= 8î -3j -6k are perpendicular.

23. Encontrar el valor de "x" de forma que λ 12i + x) + 6E y B-81-3)-6E sean perpendiculares

Answer 1

Since,$\underset{x}{\rightarrow}$.$\underset{y}{\rightarrow}$ = |$\underset{x}{\rightarrow}$|*|$\underset{y}{\rightarrow}$|*Cos($\Theta$)

here $\Theta$ = angle between vectors

if vectors are perpendicular then $\Theta$ = 90 deg

So, Cos($\Theta$) = 0

then, $\underset{x}{\rightarrow}$ .$\underset{y}{\rightarrow}$ = 0

therefore two vectors are perpendicular if there dot product is equals to zero,

here, $\underset{A}{\rightarrow}$ = 12 i + x j + 6 k

$\underset{B}{\rightarrow}$ = 8 i - 3 j - 6 k

Now, $\underset{A}{\rightarrow}$ .$\underset{B}{\rightarrow}$ = (12 i + x j + 6 k).(8 i - 3 j - 6 k)

also i.i = j.j = k.k = 1

i.j = j.k = k.i = 0

So,$\underset{A}{\rightarrow}$.$\underset{B}{\rightarrow}$ = 12*8 + x*(-3) + 6*(-6)

$\underset{A}{\rightarrow}$.$\underset{B}{\rightarrow}$ = 96 - 3x - 36

$\underset{A}{\rightarrow}$.$\underset{B}{\rightarrow}$ = 60 - 3x

for perpendicular $\underset{A}{\rightarrow}$ .$\underset{B}{\rightarrow}$ = 0

60 - 3x = 0

3x = 60

x = 20

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