Question

If the angle between the vectors a and b having direction ratios 1,2,1 and 1,3k,1 is pi/4, find k.

Answer 1

I hope it will be answer of your question

Answer 2

The value of k is $\frac{-4}{3}+\sqrt{2}$ and $\frac{-4}{3}-\sqrt{2}$.

• The angle between the vectors a and b having direction ratios 1,2,1 and 1,3k,1 is π/4.

• We have to find the value of k.

• The angle between the vectors having direction ratios $l_1,l_2,l_3$ and $m_1,m_2,m_3$ is given by

cosθ = $\frac{l_1m_1+l_2m_2+l_3m_3}{\sqrt{l_1^2+l_2^2+l_3^2}.\sqrt{m_1^2+m_2^2+m_3^2}}$

• Now substituting the given values in the equation , we get

cos(π/4) = $\frac{1.1+2.3k+1.1}{\sqrt{1+4+1}.\sqrt{1+9k^2+1}}$

$\frac{1}{\sqrt{2}}$ = $\frac{2+6k}{\sqrt{6}.\sqrt{2+9k^2}}$

$\sqrt{3}.\sqrt{2+9k^2}$ = 2+6k

• Squaring both sides , we get

3(2+$9k^2$) = (2+6k)²

6+27k² = 4+24k + 36k²

9k² + 24k - 2 = 0

• The above given equation is quadratic in k. We apply the formula method,

k = $\frac{-24+\sqrt{24^2+72}}{18}$ or k = $\frac{-24-\sqrt{24^2+72}}{18}$

k = $\frac{-24+\sqrt{576+72}}{18}$ or k = $\frac{-24-\sqrt{576+72}}{18}$

k = $\frac{-24+\sqrt{648}}{18}$ or k = $\frac{-24-\sqrt{648}}{18}$

k = $\frac{-24+18\sqrt{2}}{18}$ or k = $\frac{-24-18\sqrt{2}}{18}$

k = $\frac{-4+3\sqrt{2}}{3}$ or k = $\frac{-4-3\sqrt{2}}{3}$

k = $\frac{-4}{3}+\sqrt{2}$ or k = $\frac{-4}{3}-\sqrt{2}$