Question

If $\vec a=2\hat i+2\hat j+3 \hat k,\vec b= -\hat i+2\hat j+ \hat k\ and\ \vec c=3\hat i+ \hat j$are such that $\vec a+\lambda\vec b$ is perpendicular to $\vec c$ , then find the value of λ.

Answer 1

Solution:

Given that

If
$\vec a=2\hat i+2\hat j+3 \hat k,\\\\\vec b= -\hat i+2\hat j+ \hat k\ \\\\\ \vec c=3\hat i+ \hat j$

Also given that
$\vec a+\lambda\vec b$

is perpendicular to vector C.

$\vec a +\lambda\vec b = (2\hat i+2\hat j+3 \hat k) + \lambda(-\hat i+2\hat j+ \hat k) \\ \\ \vec a +\lambda\vec b = (2 - \lambda) \hat i + (2 + 2\lambda) \hat j + (3 + \lambda) \hat k$

If two Vectors are perpendicular than

$3.(2 - \lambda) + 1.(2 + 2\lambda) + 0.(3 + \lambda) = 0 \\ \\ 6 - 3\lambda + 2 + 2\lambda = 0 \\ \\ - \lambda = - 8 \\ \\ \lambda = 8$
Hope it helps you.