Multiple choice

A straight line passes through the point $(1,1,1)$ makes an angle $60^0$ with the positive direction of z-axis and the cosine of the angles made by it with the positive directions of the y-axis and x-axis are in the ratio $\sqrt{3}:1$. What is the acute angle between two possible positions of the line ?

  1. $90^0$
  2. $60^0$
  3. $45^0$
  4. $30^0$
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A Correct answer
Explanation

Let the direction cosines be (l, m, n). Given n = cos(60) = 1/2. Since l^2 + m^2 + n^2 = 1, l^2 + m^2 = 3/4. Given m/l = sqrt(3), so m = sqrt(3)l. Substituting, l^2 + 3l^2 = 3/4, 4l^2 = 3/4, l^2 = 3/16, l = sqrt(3)/4, m = 3/4. Two possible lines have direction ratios (sqrt(3)/4, 3/4, 1/2) and (-sqrt(3)/4, 3/4, 1/2). The dot product gives cos(theta) = (-3/16 + 9/16 + 1/4) = 0. Thus, the angle is 90 degrees.

AI explanation

Using the direction cosines property, l squared + m squared + n squared = 1. Given the angle with the z-axis is 60 degrees, n = cos 60 degrees = 0.5. The ratio of cosines for the x and y axes is given as square root of 3 to 1, meaning m = l times square root of 3. Substituting into the sum of squares formula gives l squared + 3 l squared + 0.25 = 1, so l squared = 3/16 and l is either square root of 3 divided by 4 or negative square root of 3 divided by 4. The two possible direction cosine vectors are (square root of 3 over 4, 3/4, 1/2) and (negative square root of 3 over 4, 3/4, 1/2), and their dot product is zero, meaning the angle between the two lines is 90 degrees.

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