CUET Mathematics 2024 - The direction cosines of the line which is perpendicular to the lines with direction ratios 1,-2,-2 and 0,2,1 are : | PYQs + Solutions

CUET Mathematics 2024

Algebra

Vector Algebra

Easy

The direction cosines of the line which is perpendicular to the lines with direction ratios $1,-2,-2$ and $0,2,1$ are :

\frac{2}{3},-\frac{1}{3}, \frac{2}{3}

-\frac{2}{3},-\frac{1}{3}, \frac{2}{3}

\frac{2}{3},-\frac{1}{3},-\frac{2}{3}

\frac{2}{3}, \frac{1}{3}, \frac{2}{3}

✅ Correct Option: 1

The two given lines have direction ratios:

\newline

- Line 1:

(1, -2, -2)

\newline

- Line 2:

(0, 2, 1)

When two vectors are given, their cross product gives a vector perpendicular to both.

Let's calculate the cross product

(1, -2, -2) \times (0, 2, 1)

Using the formula

\vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)

= ((-2)(1) - (-2)(2), (-2)(0) - (1)(1), (1)(2) - (-2)(0))

= (-2 - (-4), 0 - 1, 2 - 0)

= (2, -1, 2)

Direction cosines are normalized direction ratios. To normalize, divide each component by the magnitude.

Magnitude:

\|(2, -1, 2)\| = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{9} = 3

Therefore, the direction cosines are:

\left(\dfrac{2}{3}, -\dfrac{1}{3}, \dfrac{2}{3}\right)