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Line perpendicular to two given lines (direction = cross product of their directions)

This type of question, asked 10 times: 2021 Paper 4, 11 Apr 2023, 30 Jan 2024, 31 Jan 2024, 29 Jan 2025, 7 Apr 2025, 24 Jan 2026 S1, 2 Apr 2026 S1, 6 Apr 2026 S1, 8 Apr 2026 S2. All 11 questions below, oldest first.

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  1. Question 1

    2021 Paper 4

    85
    A line ' l\mathcal{l} ' passing through origin is perpendicular to the lines
    l1:r→=(3+t)i^+(−1+2t)j^+(4+2t)k^l2:r→=(3+2s)i^+(3+2s)j^+(2+s)k^\begin{matrix} & \mathcal{l}_{1}:\overrightarrow{r}= (3 + t)\widehat{i}+ ( - 1 + 2t)\widehat{j}+ (4 + 2t)\widehat{k} \\ & \mathcal{l}_{2}:\overrightarrow{r}= (3 + 2s)\widehat{i}+ (3 + 2s)\widehat{j}+ (2 + s)\widehat{k} \end{matrix}
    If the co-ordinates of the point in the first octant on ' l2\mathcal{l}_{2} ' at a distance of 17\sqrt{17} from the point of intersection of ' l\mathcal{l} ' and ' l1\mathcal{l}_{1} ' are (a,b,c)(a,b,c), then 18(a+b+c)18(a + b + c) is equal to

  2. Question 2

    11 Apr 2023

    85
    Let a line ll pass through the origin and be perpendicular to the lines
    l1:r→=(i^−11j^−7k^)+λ(i^+2j^+3k^),λ∈Rl_{1}:\overrightarrow{r}= (\widehat{i}- 11\widehat{j}- 7\widehat{k}) + \lambda(\widehat{i}+ 2\widehat{j}+ 3\widehat{k}),\lambda \in R
    and l2:r→=(−i^+k^)+μ(2i^+2j^+k^),μ∈Rl_{2}:\overrightarrow{r}= ( -\widehat{i}+\widehat{k}) + \mu(2\widehat{i}+ 2\widehat{j}+\widehat{k}),\mu \in R. If PP is the point of intersection of ll and l1l_{1}, and Q(αQ(\alpha ,β,γ),\beta,\gamma) is the foot of perpendicular from PP on l2l_{2}, then 9(α+β+γ)9(\alpha + \beta + \gamma) is equal to

  3. Question 3

    30 Jan 2024

    163
    Let L1:r→=(i^−j^+2k^)+λ(i^−j^+2k^),λ∈RL_{1}:\overrightarrow{r}= (\widehat{i}-\widehat{j}+ 2\widehat{k}) + \lambda(\widehat{i}-\widehat{j}+ 2\widehat{k}),\lambda \in R L2:r→=(j^−k^)+μ(3i^+j^+pk^),μ∈RL_{2}:\overrightarrow{r}= (\widehat{j}-\widehat{k}) + \mu(3\widehat{i}+\widehat{j}+ p\widehat{k}),\mu \in R and L3:r→=δ(li^+mj^+nk^)δ∈RL_{3}:\overrightarrow{r}= \delta(\mathcal{l}\widehat{i}+ m\widehat{j}+ n\widehat{k})\delta \in R Be three lines such that L1L_{1} is perpendicular to L2L_{2} and L3L_{3} is perpendicular to both L1L_{1} and L2L_{2}. Then the point which lies on L3L_{3} is

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  4. Question 4

    31 Jan 2024

    74
    The distance of the point Q(0,2,−2)Q(0,2, - 2) form the line passing through the point P(5,−4,3)P(5, - 4,3) and perpendicular to the lines r→=(−3i^+2k^)+\overrightarrow{r}= ( - 3\widehat{i}+ 2\widehat{k}) + λ(2i^+3j^+5k^), λ∈R\lambda(2\widehat{i}+ 3\widehat{j}+ 5\widehat{k}),\ \lambda\in\mathbb{R} and  r→=(i^−2j^+k^)+\ \overrightarrow{r}= (\widehat{i}- 2\widehat{j}+\widehat{k}) + μ(−i^+3j^+2k^),μ∈R\mu( -\widehat{i}+ 3\widehat{j}+ 2\widehat{k}),\mu\in\mathbb{R} is

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  5. Question 5

    29 Jan 2025

    131
    Let a straight line LL pass through the point P(2,−1,3)P(2, - 1,3) and be perpendicular to the lines x−12=y+11=z−3−2\frac{x - 1}{2}=\frac{y + 1}{1}=\frac{z - 3}{- 2} and x−31=y−23=z+24\frac{x - 3}{1}=\frac{y - 2}{3}=\frac{z + 2}{4}. If the line L intersects the yz -plane at the point Q , then the distance between the points P and Q is :

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  6. Question 6

    29 Jan 2025

    67
    Let L1:x−11=y−2−1=z−12L_{1}:\frac{x - 1}{1}=\frac{y - 2}{- 1}=\frac{z - 1}{2} and L2:x+1−1=y−22=z1L_{2}:\frac{x + 1}{- 1}=\frac{y - 2}{2}=\frac{z}{1} be two lines. Let L3L_{3} be a line passing through the point (α,β,γ)(\alpha,\beta,\gamma) and be perpendicular to both L1L_{1} and L2L_{2}. If L3L_{3} intersects L1L_{1}, then ∣5α−11β−8γ∣|5\alpha - 11\beta - 8\gamma| equals :

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  7. Question 7

    7 Apr 2025

    139
    If the equation of the line passing through the point (0,−12,0)\left( 0, -\frac{1}{2},0 \right) and perpendicular to the lines r→=λ(i^+aj^+bk^)\overrightarrow{r}=\lambda(\widehat{i}+ a\widehat{j}+ b\widehat{k}) and r→=(i^−j^−6k^)+μ(−bi^+aj^+5k^)\overrightarrow{r}= (\widehat{i}-\widehat{j}- 6\widehat{k}) +\mu( - b\widehat{i}+ a\widehat{j}+ 5\widehat{k}) is x−1−2=y+4 d=z−c−4\frac{x - 1}{- 2}=\frac{y + 4}{\text{ }d}=\frac{z - c}{- 4}, then a+b+c+da + b + c + d is equal to:

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  8. Question 8

    24 Jan 2026 S1

    72
    Let a line L passing through the point P(1,1,1)P(1,1,1) be perpendicular to the lines x−44=y−11=z−11\frac{x - 4}{4} = \frac{y - 1}{1} = \frac{z - 1}{1} and x−171=y−711=z0\frac{x - 17}{1} = \frac{y - 71}{1} = \frac{z}{0}. Let the line L intersect the yz -plane at the point Q . Another line parallel to L and passing through the point S(1,0,−1)S(1,0, - 1) intersects the yz -plane at the point R . Then the square of the area of the parallelogram PQRS is equal to ____\_\_\_\_ .

  9. Question 9

    2 Apr 2026 S1

    65
    Let a line L passing through the point (1,1,1)(1,1,1) be perpendicular to both the vectors 2i^+2j^+k^2\widehat{i}+ 2\widehat{j}+\widehat{k} and i^+2j^+k^\widehat{i}+ 2\widehat{j}+\widehat{k}. If P(a,b,c)P(a,b,c) is the foot of perpendicular from the origin on the line L , then the value of 34(a+b+c)34(a + b + c) is :

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  10. Question 10

    6 Apr 2026 S1

    66
    Let a line L be perpendicular to both the lines L1:x+13=y+35=z+57L_{1}:\frac{x + 1}{3}=\frac{y + 3}{5}=\frac{z + 5}{7} and L2:x−21=y−44=z−67L_{2}:\frac{x - 2}{1}=\frac{y - 4}{4}=\frac{z - 6}{7}. If θ\theta is the acute angle between the lines L and L3:x−872=y−471=z2L_{3}:\frac{x -\frac{8}{7}}{2}=\frac{y -\frac{4}{7}}{1}=\frac{z}{2}, then tan⁡θ\tan\theta is equal to:

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  11. Question 11

    8 Apr 2026 S2

    73
    Let a line L1L_{1} pass through the origin and be perpendicular to the lines L2:r→=(3+t)i^+(2t−1)j^+(2t+4)k^L_{2}:\overrightarrow{r} = (3 + t)\widehat{i} + (2t - 1)\widehat{j} + (2t + 4)\widehat{k} and L3:r→=(3+2s)i^+(3+2s)j^+(2+s)k^,tL_{3}:\overrightarrow{r} = (3 + 2s)\widehat{i} + (3 + 2s)\widehat{j} + (2 + s)\widehat{k},t, s∈Rs \in R. If (a,b,c),a∈Z(a,b,c),a \in Z, is the point on L3L_{3} at a distance of 17\sqrt{17} from the point of intersection of L1L_{1} and L2L_{2}, then (a+b+c)2(a + b + c)^{2} is equal to ____\_\_\_\_.