Multiple choice

The angle between the lines whose cosines are $\left(\dfrac{-\sqrt{3}}{4},\dfrac{1}{4},\dfrac{-\sqrt{3}}{2}\right)$ and $\left(\dfrac{-\sqrt{3}}{4},\dfrac{1}{4},\dfrac{-\sqrt{3}}{2}\right)$, is given by ___________.

  1. $30^o$
  2. $0^o$
  3. $45^o$
  4. $60^o$
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B Correct answer
Explanation

The angle between two lines with direction cosines (l1, m1, n1) and (l2, m2, n2) is given by cos(theta) = l1l2 + m1m2 + n1n2. Since the vectors are identical, the dot product is 1, so cos(theta) = 1, meaning theta = 0 degrees.

AI explanation

The angle between two lines is determined by their direction cosines. The formula for the angle theta between two lines with direction cosines (l1, m1, n1) and (l2, m2, n2) is cos theta = l1l2 + m1m2 + n1n2. Because the direction cosines for both lines are identical, cos theta = (-square root of 3/4)^2 + (1/4)^2 + (-square root of 3/2)^2 = 3/16 + 1/16 + 12/16 = 1. Therefore, theta = 0 degrees.

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