Three Dimensional Geometry — Class 12 MCQs with Answers

Three Dimensional Geometry — important questions & MCQs with answers (Class 12 Mathematics)

14 solved questions from this chapter's difficulty levels, each with its answer and explanation. The other 76 are timed and scored when you sign in.

1. Q1Easy

   Direction cosines of x-axis:

   A.(1,0,0)✓ Correct

   B.(0,1,0)

   C.(0,0,1)

   D.(1,1,1)

   Answer: A. (1,0,0)

   Explanation: The x-axis makes 0° with itself and 90° with the y and z axes, so its direction cosines are cos0°,cos90°,cos90° = 1,0,0 — the mixed values in the other options belong to different axes.

2. Q2Easy

   Plane ax+by+cz=d has normal vector:

   A.(b,c,a)

   B.(d,0,0)

   C.(a,b,c)✓ Correct

   D.(0,0,d)

   Answer: C. (a,b,c)

   Explanation: In ax+by+cz=d, the coefficients of x, y and z form the plane's normal vector directly, in that same order — swapping the order of the coefficients gives an incorrect direction entirely.

3. Q3Easy

   Direction cosines of a line: cos α, cos β, cos γ where α, β, γ are angles with:

   A.Any direction

   B.Origin

   C.Plane

   D.Positive x, y, z axes✓ Correct

   Answer: D. Positive x, y, z axes

   Explanation: Direction cosines are defined as the cosines of the angles a line makes with the positive x, y and z axes specifically — not with the origin or an arbitrary plane, which don't fix a unique direction.

4. Q4Easy

   Direction ratios of a line are:

   A.Lengths

   B.Same as DC

   C.Numbers proportional to direction cosines✓ Correct

   D.Magnitudes

   Answer: C. Numbers proportional to direction cosines

   Explanation: Direction ratios are any numbers proportional to the direction cosines l, m, n — for example 2,4,4 and 1,2,2 represent the same direction, unlike direction cosines, which are fixed and unique.

5. Q5Easy

   Vector equation of line through point a⃗ with direction b⃗:

   A.r⃗ · b⃗ = 0

   B.r⃗ = a⃗

   C.r⃗ = b⃗

   D.r⃗ = a⃗ + t b⃗✓ Correct

   Answer: D. r⃗ = a⃗ + t b⃗

   Explanation: A line through point a⃗ in direction b⃗ is written r⃗ = a⃗ + t b⃗, with the parameter t sweeping out every point on the line — without t, r⃗=a⃗ is just the single starting point.

6. Q6Easy

   Plane in normal form: x cos α + y cos β + z cos γ = p where p is:

   A.Perpendicular distance from origin✓ Correct

   B.x-intercept

   C.Direction cosine

   D.Slope

   Answer: A. Perpendicular distance from origin

   Explanation: In the normal form xcosα+ycosβ+zcosγ=p, cosα,cosβ,cosγ are the direction cosines of the normal and p is the plane's perpendicular distance from the origin, always taken positive.

7. Q7Medium

   Line through point a with direction b:

   A.r = a + b only

   B.r = a × b

   C.r = ab

   D.r = a + λb✓ Correct

   Answer: D. r = a + λb

   Explanation: A line's vector equation is a fixed point plus a scalar multiple of the direction vector: r=a+λb, with λ ranging over all reals — dropping λ freezes it to a single point, not a line.

8. Q8Medium

   Distance from origin to plane ax+by+cz=d:

   A.|d|/√(a²+b²+c²)✓ Correct

   B.d

   C.a+b+c

   D.|a|+|b|+|c|

   Answer: A. |d|/√(a²+b²+c²)

   Explanation: Distance from the origin to ax+by+cz=d is |d|/√(a²+b²+c²): the constant term over the magnitude of the normal — forgetting to divide by the normal's length is the usual mistake.

9. Q9Medium

   Direction cosines of line with direction ratios 1, 2, 2:

   A.(1/√3, 1/√3, 1/√3)

   B.(1, 2, 2)

   C.(1/3, 2/3, 2/3)✓ Correct

   D.(1/9, 2/9, 2/9)

   Answer: C. (1/3, 2/3, 2/3)

   Explanation: Divide each direction ratio by the magnitude √(1²+2²+2²)=√9=3 to get direction cosines 1/3, 2/3, 2/3 — leaving the ratios unnormalised, as the raw numbers themselves, isn't the same as direction cosines.

10. Q10Medium

    The direction ratios of the line joining (2, 3, 4) and (5, 7, 4) are:

    A.7, 10, 8

    B.3, 4, 0✓ Correct

    C.2, 3, 4

    D.5, 7, 4

    Answer: B. 3, 4, 0

    Explanation: The direction ratios are the differences 3, 4 and 0.

11. Q11Medium

    Vector equation of line through (1, 2, 3) parallel to î + ĵ + k̂:

    A.Cannot find

    B.r⃗ = î + ĵ + k̂

    C.r⃗ = î + 2ĵ + 3k̂

    D.r⃗ = (î + 2ĵ + 3k̂) + t(î + ĵ + k̂)✓ Correct

    Answer: D. r⃗ = (î + 2ĵ + 3k̂) + t(î + ĵ + k̂)

    Explanation: Combine the point's position vector with the direction: r⃗ = (î+2ĵ+3k̂) + t(î+ĵ+k̂) — using just the point or just the direction alone describes a single vector, not the whole line.

12. Q12Hard

    Angle between two lines with direction vectors u, v:

    A.sinθ = u·v

    B.u×v

    C.u·v / |u+v|

    D.cosθ = (u·v)/(|u||v|)✓ Correct

    Answer: D. cosθ = (u·v)/(|u||v|)

    Explanation: The angle between two lines uses the same dot-product rule as any two vectors: cosθ=(u·v)/(|u||v|); using sinθ or a plain dot product without normalising by both magnitudes gives a wrong ratio.

13. Q13Hard

    Two non-parallel, non-intersecting lines are:

    A.Skew✓ Correct

    B.Parallel

    C.Coincident

    D.Same

    Answer: A. Skew

    Explanation: Skew lines are neither parallel (direction vectors aren't proportional) nor intersecting, so no single plane contains both — that is what makes them skew rather than merely non-intersecting.

14. Q14Hard

    If a line makes angles of 60°, 60° and 45° with the axes, the sum of the squares of its direction cosines is:

    A.2

    B.3/2

    C.1✓ Correct

    D.1/2

    Answer: C. 1

    Explanation: cos²60° + cos²60° + cos²45° = 1/4 + 1/4 + 1/2 = 1, as it must be for any line.

Three Dimensional Geometry — FAQs

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