The direction ratios of the line perpendicular to the lines with direction ratios < 1, -2, -2 > and < 0, 2, 1 > are
< 2, -1, 2 >
Finding Direction Ratios of Perpendicular Lines
We are asked to find the direction ratios of a line that is perpendicular to two given lines. The direction ratios of the two given lines are $\langle 1, -2, -2 \rangle$ and $\langle 0, 2, 1 \rangle$.
If a line is perpendicular to two lines with direction ratios $\langle a_1, b_1, c_1 \rangle$ and $\langle a_2, b_2, c_2 \rangle$, then the direction ratios of the perpendicular line, $\langle a, b, c \rangle$, are proportional to the cross product of the direction vectors corresponding to the given direction ratios.
The cross product of two vectors $\mathbf{v}_1 = a_1\mathbf{i} + b_1\mathbf{j} + c_1\mathbf{k}$ and $\mathbf{v}_2 = a_2\mathbf{i} + b_2\mathbf{j} + c_2\mathbf{k}$ is given by:
$\mathbf{v}_1 \times \mathbf{v}_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix} = (b_1c_2 - b_2c_1)\mathbf{i} - (a_1c_2 - a_2c_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$
The direction ratios of the resulting vector are $\langle (b_1c_2 - b_2c_1), -(a_1c_2 - a_2c_1), (a_1b_2 - a_2b_1) \rangle$, which is equivalent to $\langle (b_1c_2 - b_2c_1), (a_2c_1 - a_1c_2), (a_1b_2 - a_2b_1) \rangle$.
Given direction ratios:
- Line 1: $\langle a_1, b_1, c_1 \rangle = \langle 1, -2, -2 \rangle$
- Line 2: $\langle a_2, b_2, c_2 \rangle = \langle 0, 2, 1 \rangle$
Let the direction ratios of the perpendicular line be $\langle a, b, c \rangle$. We can calculate these using the cross product formula:
- $a = b_1c_2 - b_2c_1 = (-2)(1) - (2)(-2) = -2 - (-4) = -2 + 4 = 2$
- $b = c_1a_2 - c_2a_1 = (-2)(0) - (1)(1) = 0 - 1 = -1$
- $c = a_1b_2 - a_2b_1 = (1)(2) - (0)(-2) = 2 - 0 = 2$
So, the direction ratios of the line perpendicular to the given lines are proportional to $\langle 2, -1, 2 \rangle$.
Now, let's compare this with the given options:
- $\langle 2, -1, 2 \rangle$
- $\langle -2, 1, 2 \rangle$
- $\langle 2, 1, -2 \rangle$
- $\langle -2, -1, -2 \rangle$
The calculated direction ratios $\langle 2, -1, 2 \rangle$ match the first option.
Conclusion
The direction ratios of the line perpendicular to the lines with direction ratios $\langle 1, -2, -2 \rangle$ and $\langle 0, 2, 1 \rangle$ are $\langle 2, -1, 2 \rangle$.
Similar Questions
A point on a line has coordinates (p + 1, p - 3, √2p) where p is any real number. What are the direction cosines of the line?
If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are the coordinates of the points on L whose distance from (1, -1, 1) is 2 units?
If l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z, then what is l 4+ m 4+ n 4equal to?
A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
A straight line with direction cosines \(\left\langle {0,\;1,\;0} \right\rangle\) is
If a line has direction ratios < a + b, b + c, c + a >, then what is the sum of the squares of its direction cosines?
Consider the following statements :
1. The direction ratios of y-axis can be <0, 4, 0>
2. The direction ratios of a line perpendicular to z-axis can be <5, 6, 0>
Which of the statements given above is/are correct?
What are the direction cosines of z-axis?
Consider the following statements:
1. A-line in space can have infinitely many direction ratios.
2. It is possible for certain lines that the sum of the squares of direction cosines can be equal to the sum of its direction cosines.
Which of the above statements is/are correct?
If α, β and γ are the angles which the vector \(\overrightarrow {OP} \) (O being the origin) makes with the positive direction of the coordinate axes, then which of the following are correct?
1. cos 2α + cos 2β = sin 2γ
2. sin 2α + sin 2β = cos 2γ
3. sin 2α + sin 2β + sin 2γ = 2
Select the correct answer using the code given below.
Important Questions from Direction ratios and Direction cosines
A point on a line has coordinates (p + 1, p - 3, √2p) where p is any real number. What are the direction cosines of the line?
If L is the line with direction ratios < 3, -2, 6 > and passing through (1, -1, 1), then what are the coordinates of the points on L whose distance from (1, -1, 1) is 2 units?
If l, m, n are the direction cosines of the line x - 1 = 2(y + 3) = 1 - z, then what is l 4+ m 4+ n 4equal to?
A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
Which of the following is the direction cosines of z, y and x-axis?
