Q.If $\vec{a}\times\vec{b} = \vec{c}\times\vec{d}$ and $\vec{a}\times\vec{c} = \vec{b}\times\vec{d}$, then prove that $\vec{a} - \vec{d}$ is parallel to $\vec{b} - \vec{c}$. OR The four vertices of a tetrahedron are respectively $O(0,0,0)$, $A(1,2,1)$, $B(2,1,3)$ and $C(1,1,2)$. Find the volume of the tetrahedron.

Showing the 12 most recent of 15 on this concept.

1. CBSE 2026Set ANNUAL1 markMCQ
   Q.If |vector a| = 5 units then |vector a × vector a| is equal to:

   (a) 5 units

   (b) 25 units

   (c) 1 units

   (d) 0 units

   ›Reveal solutionSolution

   The cross product of any vector with itself is always the zero vector, because the angle between a vector and itself is $0$.

   For any vector $\vec a$,

   $\vec a \times \vec a = |\vec a||\vec a|\sin(0^\circ)\,\hat n = 0$

   …

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2. CBSE 2025Set E1 markMCQ
   Q.If the direction ratios of two parallel lines are $2, 7, 9$ and $6, 21, x$, then the value of $x$ is

   (a) $9$

   (b) $18$

   (c) $27$

   (d) $3$

   ›Reveal solutionSolution

   Proportional DRs: $\frac{2}{6}=\frac{7}{21}=\frac{9}{x}=\frac13\Rightarrow x=27$.

   Parallel lines have direction ratios in proportion:

   $\frac{2}{6}=\frac{7}{21}=\frac{9}{x}.$

   …

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3. CBSE 2025Set E1 markMCQ
   Q.If the direction ratios of two parallel lines are $a, b, c$ and $x, y, z$ then $az =$

   (a) $cy$

   (b) $cx$

   (c) $bz$

   (d) $ax$

   ›Reveal solutionSolution

   Proportional DRs: $\frac{a}{x}=\frac{b}{y}=\frac{c}{z}$, so cross-multiplying the first and third gives $az=cx$.

   For parallel lines the direction ratios are proportional:

   $\frac{a}{x}=\frac{b}{y}=\frac{c}{z}.$

   …

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4. CBSE 2025Set ANNUAL1 markMCQ
   Q.The value of $\lambda$ for which the vectors $3\hat{i} - 6\hat{j} + \hat{k}$ and $2\hat{i} - 4\hat{j} + \lambda\hat{k}$ are parallel is

   (a) $\dfrac{2}{3}$

   (b) $\dfrac{3}{2}$

   (c) $\dfrac{5}{2}$

   (d) $\dfrac{2}{5}$

   ›Reveal solutionSolution

   Parallel vectors have proportional components; match the third component's ratio to the other two.

   For $3\hat i - 6\hat j + \hat k$ and $2\hat i - 4\hat j + \lambda\hat k$ to be parallel, corresponding components must be in the same ratio:

   $\dfrac{3}{2} = \dfrac{-6}{-4} = \dfrac{1}{\lambda}$

   …

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5. CBSE 2025Set ANNUAL1 markMCQ
   Q.If a⃗ is a non-zero vector then |a⃗ × a⃗| is equal to:

   (a) |a⃗|

   (b) |a⃗|²

   (c) 1

   (d) 0

   ›Reveal solutionSolution

   The cross product of any vector with itself is always the zero vector, so its magnitude is $0$.

   For any non-zero vector $\vec{a}$, the angle between $\vec{a}$ and itself is $\theta = 0$. Using $\vec{a}\times\vec{a} = |\vec{a}||\vec{a}|\sin\theta\,\hat n$, and $\sin 0 = 0$, the whole expres …

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6. CBSE 2024Set A11 mark
   Q.Two lines with direction ratios 1, 3, 5 and 2, $K$, 10 are parallel then the value of $K$ is ________.

   ›Reveal solutionSolution

   Proportional direction ratios for parallel lines give $\dfrac{3}{K}=\dfrac{1}{2}$, hence $K=6$.

   Two lines are parallel iff their direction ratios are proportional. With ratios $1,3,5$ and $2,K,10$,

   $\frac{1}{2}=\frac{3}{K}=\frac{5}{10}.$ …

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7. CBSE 2024Set D1 markMCQ
   Q.If the line $\frac{x-3}{a} = \frac{y-4}{b} = \frac{z-5}{c}$ is parallel to the line $\frac{x}{5} = \frac{y}{3} = \frac{z}{2}$, then

   (a) $5a + 3b + 2c = 0$

   (b) $\frac{a}{5} = \frac{b}{3} = \frac{c}{2}$

   (c) $5a = 3b = 2c$

   (d) none of these

   ›Reveal solutionSolution

   Parallel lines $\Rightarrow$ direction ratios proportional: $\frac{a}{5} = \frac{b}{3} = \frac{c}{2}$.

   The first line has direction ratios $(a, b, c)$ and the second has $(5, 3, 2)$. Two lines are parallel iff their direction ratios are proportional:

   $\frac{a}{5} = \frac{b}{3} = \frac{c}{2}.$ …

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8. CBSE 2023Set 65/3/11 markMCQ
   Q.(Assertion-Reason) Assertion (A) : A line through the points $(4, 7, 8)$ and $(2, 3, 4)$ is parallel to a line through the points $(-1, -2, 1)$ and $(1, 2, 5)$. Reason (R) : Lines $\vec{r} = \vec{a_1} + \lambda \vec{b_1}$ and $\vec{r} = \vec{a_2} + \mu \vec{b_2}$ are parallel if $\vec{b_1} \cdot \vec{b_2} = 0$.

   (a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

   (b) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

   (c) Assertion (A) is true and Reason (R) is false.

   (d) Assertion (A) is false and Reason (R) is true.

   ›Reveal solutionSolution

   Two lines are parallel when their direction vectors are scalar multiples of each other. The direction vectors of both given line pairs are $\langle -2, -4, -4 \rangle$ and $\langle 2, 4, 4 \rangle$, which are scalar multiples, so Assertion (A) is true. Reason (R) is false because the condition for parallel lines is $\vec{b_1} \times \vec{b_2} = \vec{0}$, not $\vec{b_1} \cdot \vec{b_2} = 0$. The correct option is (c).

   Concept First: What Makes Two Lines Parallel?

   In 3D geometry, two lines are parallel if their direction vectors are scalar multiples of each other. That is, if one direction vector can be written as $k$ times the other, for some non-zero scalar $k$. This is equivalent to saying their cross product is the zero vector: $\vec{b_1} \times \vec{b_2} = \vec{0}$.

   The dot product condition $\vec{b_1} \cdot \vec{b_2} = 0$ means the vectors are perpendicular, not parallel. That's a classic trap — Reason (R) states exactly this wrong condition.

   Let's check the Assertion first.

   ---

   Step-by-Step Solution

   1. Find the direction vector of the first line.

   The line passes through $(4, 7, 8)$ and $(2, 3, 4)$. The direction vector is the difference between these points:

   $\vec{b_1} = (2 - 4, \; 3 - 7, \; 4 - 8) = (-2, -4, -4)$

   2. Find the direction vector of the second line.

   The line passes through $(-1, -2, 1)$ and $(1, 2, 5)$. Its direction vector is:

   $\vec{b_2} = (1 - (-1), \; 2 - (-2), \; 5 - 1) = (2, 4, 4)$

   3. Check if they are parallel.

   Observe that $\vec{b_2} = -1 \cdot \vec{b_1}$:

   $(2, 4, 4) = -1 \cdot (-2, -4, -4)$

   Since one is a scalar multiple of the other, the two lines are parallel. Assertion (A) is true.

   Tip

   You don't always need to compute the scalar factor explicitly. Just check if the ratios of corresponding components are equal: $\frac{2}{-2} = \frac{4}{-4} = \frac{4}{-4} = -1$. If all three ratios match, the vectors are parallel. …

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9. CBSE 2023Set E1 markMCQ
   Q.If the direction ratios of two parallel lines are $x,5,3$ and $20,10,6$ then the value of $x$ is

   (a) $10$

   (b) $5$

   (c) $3$

   (d) $40$

   ›Reveal solutionSolution

   Proportional direction ratios give $x=10$.

   Parallel lines have proportional direction ratios, so

   $\frac{x}{20}=\frac{5}{10}=\frac{3}{6}.$

   …

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10. CBSE 2023Set E1 markMCQ
    Q.If the direction ratios of two parallel lines are $a_{1},b_{1},c_{1}$ and $a_{2},b_{2},c_{2}$ then $\dfrac{a_{1}c_{2}}{a_{2}} =$

    (a) $b_{1}$

    (b) $b_{2}$

    (c) $b_{3}$

    (d) $c_{1}$

    ›Reveal solutionSolution

    From proportional direction ratios, $\dfrac{a_1c_2}{a_2}=c_1$.

    For parallel lines the direction ratios are proportional:

    $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}.$

    …

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11. CBSE 2022Set ANNUAL1 markMCQ
    Q.$3\vec a \times 4\vec a = ?$

    (a) $0$

    (b) $12$

    (c) $12\vec a$

    (d) $\vec 0$

    ›Reveal solutionSolution

    The cross product of any vector with itself (or a scalar multiple of itself) is the zero vector.

    $3\vec a\times4\vec a=12(\vec a\times\vec a)$.

    …

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12. CBSE 2020Set ANNUAL1 mark
    Q.For what value of $p$ the vectors $3\hat{i}+2\hat{j}+9\hat{k}$ and $\hat{i}+p\hat{j}+3\hat{k}$ are parallel.

    ›Reveal solutionSolution

    Two vectors are parallel exactly when their corresponding components are proportional; matching ratios gives $p$.

    Given vectors $3\hat{i}+2\hat{j}+9\hat{k}$ and $\hat{i}+p\hat{j}+3\hat{k}$ are parallel, so their components must be proportional:

    $\frac{3}{1} = \frac{2}{p} = \frac{9}{3}$

    …

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