Question

If vector A = { i + cj + 5k and vector B = 2î + j - k}
are perpendicular, then calculate the value of c.
​

Answer 1

for two vectors to be perpendicular their dot product has to be zero

so A•B=0

A•B=(1*2)+(c*1)+(5*(-1))

=2+c-5=c-3=0

c=3

Answer 2

Answer:

The value of c is equal to 3.

Explanation:

We have given, the vector, $\vec A =\hat i+c\hat j+5\hat k$

The vector, $\vec B =\hat 2i+\hat j-\hat k$

We know that the dot product of two vectors is given by:

$\vec A.\vec B =|\vec A| |\vec B|cos\theta$

where θ is the angle between two vectors A and B.

We have given, the two vectors $\vec A \; \; and \; \; \vec B$ are perpendicular to each other. Therefore, θ =90°.

We know, the value of cos90° = 0.

$\vec A.\vec B =0$                                                                         ................(1)

From here, we can say this if two vectors are perpendicular to each other then, their dot product will be equal to zero.

Substitute the value of A and B vectors in equation (1).

$(\hat i+c\hat j+5\hat k).(\hat 2i+\hat j-\hat k)=0$

$(\hat i).(\hat 2i) +(\hat cj).(\hat j)+(\hat 5k).(\hat -k)=0$

$2+c-5=0$

$c-3=0$

$c=3$

Therefore, the value of c is equal to 3.

Hence, the vector A becomes  $\vec A =\hat i+3\hat j+5\hat k$.