Let A(-1,2, -3), B(5, 0, -6), C(0, 4, -1) be the threepoints.Then the direction cosines of the internal bisector of the angle BAC are proportional toa)(25,8,5)b)(-11,20,23)c)(7,-6,-9)d)none of the aboveCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Proof in short : verify that
AB = 7 and AC = 3

where θ is the angle between AB (with d.c.'s [l₁, m₁, n₁]) and AC (with d.c.’s [l₂, m₂, n₂)
∴ the d.c.’s of the internal bisector are proprotional to l₁+l₂, m₁+m₂, n₁+n₂
or proportional to

or proportional to

or proportional to 25, 8, 5

Answer

Given Points:
- A(-1, 2, -3)
- B(5, 0, -6)
- C(0, 4, -1)

Finding the direction cosines of the internal bisector of angle BAC:
- First, we need to find the vectors representing the sides of the angle BAC.
- Let's calculate the vectors:
- Vector BA = A - B = (-1-5)i + (2-0)j + (-3+6)k = -6i + 2j + 3k
- Vector BC = C - B = (0-5)i + (4-0)j + (-1+6)k = -5i + 4j + 5k
- Next, find the bisector vector:
- Internal bisector = (BA/|BA| + BC/|BC|) / sqrt((BA/|BA| + BC/|BC|)^2)

Calculating the direction cosines:
- The direction cosines of a vector are the cosines of the angles between the vector and the positive x, y, and z axes.
- Let the direction cosines of the internal bisector vector be (l, m, n).
- Normalize the vector to find the direction cosines:
- l = x-component / magnitude
- m = y-component / magnitude
- n = z-component / magnitude
- Calculate the direction cosines using the normalized internal bisector vector.

Comparing the results:
- Once you have the direction cosines, compare them to the options provided.
- The correct answer is the option where the direction cosines are proportional to the given values.
- In this case, the correct answer is option 'A' with direction cosines proportional to (25, 8, 5).

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