For the following two (02) items:
A plane $P$ is parallel to the line having direction ratios $\langle 1, 3, 2 \rangle$ and contains the line of intersection of the planes $6x + 4y - 5z = 2$ and $x - 2y + 3z = 0$ .

Which of the following are the direction ratios of the line of intersection of the given planes?

Question

For the following two (02) items:
A plane

P

is parallel to the line having direction ratios

\langle 1, 3, 2 \rangle

and contains the line of intersection of the planes

6x + 4y - 5z = 2

and

x - 2y + 3z = 0

.

Which of the following are the direction ratios of the line of intersection of the given planes?

Quantrex Academy · Accepted Answer

Correct answer: B. 2, -23, -16

The normal vectors of the given planes are

\vec{n}_1 = \langle 6, 4, -5 \rangle

and

\vec{n}_2 = \langle 1, -2, 3 \rangle

.

The direction vector

\vec{d}

of the line of intersection of the planes is given by the cross product of their normal vectors:

\vec{d} = \vec{n}_1 \times \vec{n}_2 = \begin{vmatrix} \hat{i} \text{ and } \hat{j} \text{ and } \hat{k} \\ 6 \text{ and } 4 \text{ and } -5 \\ 1 \text{ and } -2 \text{ and } 3 \end{vmatrix}

\vec{d} = \hat{i}(12 - 10) - \hat{j}(18 + 5) + \hat{k}(-12 - 4)

\vec{d} = 2\hat{i} - 23\hat{j} - 16\hat{k}

The direction ratios of the line of intersection are

\langle 2, -23, -16 \rangle

.

Answer:

\langle 2, -23, -16 \rangle

For the following two (02) items: A plane P is parallel to the line...