Miscellaneous Exercise answers: Three Dimensional Geometry

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1. Perpendicularity only needs the dot product of the two direction-ratio triples to be zero — no lengths are needed for that.
2. Dot product: a(b−c) + b(c−a) + c(a−b).
3. Expand: ab − ac + bc − ab + ac − bc.
4. The ab terms cancel, the ac terms cancel, and the bc terms cancel, leaving 0.
5. This holds for every choice of a, b, c, not just special ones, so the two lines are always perpendicular.
6. The angle between them is 90°.

Answer90° (the lines are perpendicular for every choice of a, b, c)

Watch this explained “Square for every choice of three letters”, 14:04 into The angle between two lines, from cosines or from ratios

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1. The line passes through the origin, so its anchor point is (0, 0, 0).
2. A line parallel to the x-axis has the same direction as the x-axis itself: direction ratios 1, 0, 0.
3. Vector form: r⃗ = λî.
4. Cartesian form: x/1 = y/0 = z/0 — read as y = 0 and z = 0, with x free to take any value.

Answerr⃗ = λî; as conditions, y = 0 and z = 0 (x runs free) — the x-axis itself.

Watch this explained “A nought underneath”, 15:58 into One point plus one direction fixes a line, in vector form and in the symmetric Cartesian form the parameter eliminates to

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1. Given: (x − 1)/(−3) = (y − 2)/(2k) = (z − 3)/2 and (x − 1)/(3k) = (y − 1)/1 = (z − 6)/(−5).
2. Read the direction ratios off the denominators: first line (−3, 2k, 2), second line (3k, 1, −5).
3. Perpendicular means the dot product is zero: (−3)(3k) + (2k)(1) + (2)(−5) = 0.
4. −9k + 2k − 10 = 0.
5. −7k = 10, so k = −10/7.

Answerk = −10/7

Watch this explained “Choosing an unknown to make them square”, 12:21 into The angle between two lines, from cosines or from ratios

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1. Given: r⃗ = 6î + 2ĵ + 2k̂ + λ(î − 2ĵ + 2k̂) and r⃗ = −4î − k̂ + μ(3î − 2ĵ − 2k̂).
2. Anchors: a⃗₁ = (6, 2, 2), a⃗₂ = (−4, 0, −1). Directions: b⃗₁ = (1, −2, 2), b⃗₂ = (3, −2, −2).
3. Anchor gap a⃗₂ − a⃗₁ = (−10, −2, −3).
4. Cross the two directions: b⃗₁ × b⃗₂ = ((−2)(−2)−(2)(−2), (2)(3)−(1)(−2), (1)(−2)−(−2)(3)) = (8, 8, 4).
5. Length of the cross product: √(64+64+16) = √144 = 12.
6. Numerator: (−10,−2,−3)·(8,8,4) = −80−16−12 = −108, and |−108| = 108.
7. Shortest distance = 108/12 = 9.

Answer9 units

Watch this explained “The awkward one, and a whole number”, 24:34 into Skew lines, what shortest distance can mean when two lines never meet, and computing it in both the skew and parallel cases

Open NCERT p. 391Matches NCERT’s answer

1. The required line needs a direction at right angles to both given lines at once — cross their two direction vectors.
2. First given line's direction: (3, −16, 7). Second given line's direction: (3, 8, −5).
3. Cross product: ((−16)(−5)−(7)(8), (7)(3)−(3)(−5), (3)(8)−(−16)(3)) = (80−56, 21+15, 24+48) = (24, 36, 72).
4. Every entry has a common factor of 12, so divide through: (2, 3, 6) is an equally good direction.
5. The required line passes through (1, 2, −4) with direction (2, 3, 6).
6. Vector form: r⃗ = (î + 2ĵ − 4k̂) + λ(2î + 3ĵ + 6k̂).

Answerr⃗ = (î + 2ĵ − 4k̂) + λ(2î + 3ĵ + 6k̂)

Watch this explained “A direction that has to be made”, 6:38 into One point plus one direction fixes a line, in vector form and in the symmetric Cartesian form the parameter eliminates to