If \langle l,m,n \rangle are direction cosines of S, then what is the value of 43(l^{2}-m^{2}-n^{2})?

Explanation

The direction ratios of S are \langle -7, 6, 1 \rangle. The magnitude is \sqrt{(-7)^2 + 6^2 + 1^2} = \sqrt{86}. The direction cosines are l = \frac{-7}{\sqrt{86}}, m = \frac{6}{\sqrt{86}}, n = \frac{1}{\sqrt{86}}. The value required is 43(l^2 - m^2 - n^2) = 43\left(\frac{49}{86} - \frac{36}{86} - \frac{1}{86}\right) = 43\left(\frac{12}{86}\right) = 43 \times \frac{6}{43} = 6.

Related questions on 3D Geometry

• Consider the points A(2,4,6), B(-2,-4,-2), C(4,6,4) and D(8,14,12). Which of the following statements is/are correct? 1. The points ...
• Consider the equation of a sphere x^{2}+y^{2}+z^{2}-4x-6y-8z-16=0. Which of the following statements is/are correct? 1. z-axis is tangent ...
• A plane cuts intercepts 2, 2, 1 on the coordinate axes. What are the direction cosines of the normal to the plane?
• Consider the following statements : 1. The direction ratios of y-axis can be \langle 0, 4, 0 \rangle 2. The direction ratios of a line <st...
• If L is the line with direction ratios \lt 3,-2, 6\gt and passing through (1,-1,1), then what are the coordinates of the points on $L...