Vector Algebra PYQs - Last 5 Years

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Practice 298 MHT CET Mathematics questions from Vector Algebra. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

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Last 5 Years Vector Algebra Questions

Showing 48 of 298 filtered questions.

1Vector Algebra

Let \(\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}\) and \(\bar{b}=\hat{i}+\hat{j}\). If \(\bar{c}\) is a vector such that \(\bar{a} \cdot \bar{c}=|\bar{c}|,|\bar{c}-\bar{a}|=2 \sqrt{2}\) and the angle between \(\bar{a} \times \bar{b}\) and $$\bar{c...

2Vector Algebra

If \(\theta\) is angle between the vectors \(\bar{a}\) and \(\bar{b}\) where \(|\bar{a}|=4,|\bar{b}|=3\) and \(\theta \in\left(\frac{\pi}{4}, \frac{\pi}{3}\right)\), then $$|(\bar{a}-\bar{b}) \times(\bar{a}+\bar{b})|^2+4(\bar{a} \cdot \bar{...

3Vector Algebra

A vector \(\bar{a}\) has components 1 and \(2 p\) with respect to a rectangular Cartesian system. This system is rotated through a certain angle about origin in the counter clock wise sense. If, with respect to the new system, \(\bar{a}\) h...

4Vector Algebra

If \(\bar{a}=2 \hat{i}+3 \hat{j}-4 \hat{k}\) and \(\bar{b}=\hat{i}-\hat{j}-\hat{k}\), then the projection of \(\bar{b}\) in the direction of \(\bar{a}\) is

5Vector Algebra

If \(\hat{\mathrm{a}}\) and \(\hat{\mathrm{b}}\) are unit vectors and \(\overline{\mathrm{c}}=\hat{\mathrm{b}}-(\hat{\mathrm{a}} \times \overline{\mathrm{c}})\), then minimum value of \([\hat{a} \hat{b} \bar{c}]\) is

6Vector Algebra

The scalar product of vectors \(\overline{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and a unit vector along the sum of vectors \(\bar{b}=2 \hat{i}-4 \hat{j}+5 \hat{k}\) and $$\bar{c}=\lambda \hat{i}+2 \hat{j}-3 \hat...

7Vector Algebra

If \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) are non-coplanar unit vectors such that \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}\), then the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is

8Vector Algebra

\(\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}\) are three vectors. For a vector $$\mathb...

9Vector Algebra

If \(\mathbf{a}=\frac{1}{\sqrt{10}}(3 \hat{\mathbf{i}}+\hat{\mathbf{k}}), \mathbf{b}=\frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})\), then the value of $$(2 \mathbf{a}-\mathbf{b}) \cdot[(\mathbf{a} \times \mathbf{b})...

10Vector Algebra

If the vectors \(p \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+q \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+r \hat{\mathbf{k}}(p \neq q \neq r \neq 1)\) are coplanar, then the value ...

11Vector Algebra

\(\overline{\mathrm{u}}, \overline{\mathrm{v}}, \overline{\mathrm{w}}\) are three vectors such that \(|\overline{\mathrm{u}}|=1, |\bar{v}|=2,|\bar{w}|=3\). If the projection of \(\bar{v}\) along \(\bar{u}\) is equal to projection of $$\bar{...

12Vector Algebra

Let \(\overline{\mathrm{A}}\) be a vector parallel to line of intersection of planes \(P_1\) and \(P_2\) through origin. \(P_1\) is parallel to the vectors \(2 \hat{j}+3 \hat{k}\) and \(4 \hat{j}-3 \hat{k}\) and \(P_2\) is parallel to $$\ha...

13Vector Algebra

If the area of the triangle with vertices \((1,2,0)\), \((1,0,2)\) and \((0, x, 1)\) is \(\sqrt{6}\) square units, then the value of $x$ is

14Vector Algebra

Two adjacent sides of a parallelogram \(\mathrm{ABCD}\) are given by \(\overline{A B}=2 \hat{i}+10 \hat{j}+11 \hat{k}\) and \(\overline{\mathrm{AD}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}\). The side \(\mathrm{AD}\) is rota...

15Vector Algebra

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are three vectors, \(|\overline{\mathrm{a}}|=2,|\overline{\mathrm{b}}|=4,|\overline{\mathrm{c}}|=1, |\bar{b} \times \bar{c}|=\sqrt{15}\) and $$\bar{b}=2 \bar{c}+\lam...

16Vector Algebra

If

then \(|\overrightarrow{\mathrm{u}} \times \overrightarrow{\mathrm{v}}| \text { is }\)

17Vector Algebra

If \(\vec{a}, \vec{b}, \vec{c}\) are three non-zero vectors, no two of them are collinear, \(\vec{a}+2 \vec{b}\) is collinear with \(\vec{c}, \vec{b}+3 \vec{c}\) is collinear with \(\vec{a}\), then \(\vec{a}+2 \vec{b}\) is

18Vector Algebra

The unit vector which is orthogonal to the vector \(3 \hat{i}+2 \hat{j}+6 \hat{k}\) and coplanar with the vectors \(2 \hat{i}+\hat{j}+\hat{k}\) and \(\hat{i}+\hat{j}+\hat{k}\) is

19Vector Algebra

Two adjacent of sides parallelogram \(\mathrm{ABCD}\) are given by \(\overline{\mathrm{AB}}=2 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}\) and \(\overline{A D}=-\hat{i}+2 \hat{j}+2 \hat{k}\). The side \(A D\) is rotated by ang...

20Vector Algebra

\(A, B, C, D\) are four points in a plane with position vectors \(\bar{a}, \bar{b}, \bar{c}, \bar{d}\) respectively such that \((\bar{a}-\bar{d}) \cdot(\bar{b}-\bar{c})=(\bar{b}-\bar{d}) \cdot(\bar{c}-\bar{a})=0\). The point \(D\), then is ...

21Vector Algebra

If \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) are two unit vectors such that \(\overline{\mathrm{a}}+2 \overline{\mathrm{b}}\) and \(5 \bar{a}-4 \bar{b}\) are perpendicular to each other, then the angle between \(\bar{a}\) and ...

22Vector Algebra

Let \(\bar{a}, \bar{b}\) and \(\bar{c}\) be three unit vectors such that \(\bar{a} \times(\bar{b} \times \bar{c})=\frac{\sqrt{3}}{2}(\bar{b}+\bar{c})\). If \(\bar{b}\) is not parallel to \(\bar{c}\), then the angle between \(\bar{a}\) and $...

23Vector Algebra

If \(\quad \overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}, \quad \overline{\mathrm{b}}=2 \hat{\mathrm{j}}-\hat{\mathrm{k}} \quad\) and $$\quad \overline{\mathrm{r}} \times \overline{\mathrm{a}}=\overline{\mathrm{b}} \times \overlin...

24Vector Algebra

Let \(\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}\) and \(\bar{b}=\hat{i}+\hat{j}\). If \(\bar{c}\) is a vector such that $$\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=|\overline{\mathrm{c}}|,|\overline{\mathrm{c}}-\overline{\mathrm{a}}|=2 \s...

25Vector Algebra

If \(\bar{a}, \bar{b}, \bar{c}\) are three vectors such that $$\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}}+\overline{\mathrm{c}})+\overline{\mathrm{b}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})+\overline{\mathrm{c}} \cdot(\ov...

26Vector Algebra

If the volume of the parallelopiped is \(158 \mathrm{~cu}\). units whose coterminus edges are given by the vectors \(\bar{a}=(\hat{i}+\hat{j}+n \hat{k}), \bar{b}=2 \hat{i}+4 \hat{j}-n \hat{k}\) and \(\bar{c}=\hat{i}+n \hat{j}+3 \hat{k}\), w...

27Vector Algebra

If \(\overline{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overline{\mathrm{b}}=4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}\) and $$\overline{\mathrm{c}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\beta \hat{...

28Vector Algebra

If \(\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}\) and \(\overline{\mathrm{c}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) are such tha...

29Vector Algebra

If the area of the parallelogram with \(\bar{a}\) and \(\bar{b}\) as two adjacent sides is \(16 \mathrm{sq}\). units, then the area of the parallelogram having \(3 \overline{\mathrm{a}}+2 \overline{\mathrm{b}}\) and $$\overline{\mathrm{a}}+...

30Vector Algebra

Let \(\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \bar{b}=\hat{i}+\hat{j}\) and \(\bar{c}\) be a vector such that \(|\bar{c}-\bar{a}|=4,|(\bar{a} \times \bar{b}) \times \bar{c}|=3\) and the angle between \(\overline{\mathrm{c}}\) and $$\overline{\...

31Vector Algebra

The unit vector perpendicular to each of the vectors \(\bar{a}+\bar{b}\) and \(\bar{a}-\bar{b}\), where \(\bar{a}=\hat{i}+\hat{j}+\hat{k}\) and \(\overline{\mathrm{b}}=3 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}\) is

32Vector Algebra

If \(\bar{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \bar{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}, \bar{c}=2 \hat{i}-\hat{j}+4 \hat{k}\), then a vector \(\overline{\mathrm{d}}\) which is parallel to vector $$\overline{\mathrm{a}} \times \overline{\mathrm{b}}...

33Vector Algebra

Let \(\bar{a}, \bar{b}, \bar{c}\) be three non-zero vectors, such that no two of them are collinear and \((\bar{a} \times \bar{b}) \times \bar{c}=\frac{1}{3}|\bar{b}||\bar{c}| \bar{a}\). If \(\theta\) is the angle between the vectors $$\bar...

34Vector Algebra

If \(\bar{a}=\hat{i}+2 \hat{j}+\hat{k}, \bar{b}=\hat{i}-\hat{j}+\hat{k}, \bar{c}=\hat{i}+\hat{j}-\hat{k}\), then a vector in the plane of \(\bar{a}\) and \(\bar{b}\), whose projection on \(\overline{\mathrm{c}}\) is \(\frac{1}{\sqrt{3}}\), ...

35Vector Algebra

The vectors are \(\bar{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \bar{b}=\hat{i}+\hat{j}\). If \(\bar{c}\) is a vector such that \(\bar{a} \cdot \bar{c}=|\bar{c}|\) and \(|\bar{c}-\bar{a}|=2 \sqrt{2}\), angle between \(\bar{a} \times \bar{b}\) and $$...

36Vector Algebra

The vector projection of \(\overline{\mathrm{AB}}\) on \(\overline{\mathrm{CD}}\), where \(A \equiv(2,-3,0), B \equiv(1,-4,-2), C \equiv(4,6,8)\) and \(\mathrm{D} \equiv(7,0,10)\), is

37Vector Algebra

If \(\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overline{\mathrm{b}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}\) and \(\overline{\mathrm{c}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}\) are such tha...

38Vector Algebra

Scalar projection of the line segment joining the points \(\mathrm{A}(-2,0,3), \mathrm{B}(1,4,2)\) on the line whose direction ratios are \(6,-2,3\) is

39Vector Algebra

If the volume of tetrahedron, whose vertices are with position vectors \(\hat{i}-6 \hat{j}+10 \hat{k},-\hat{i}-3 \hat{j}+7 \hat{k}, 5 \hat{i}-\hat{j}+\lambda \hat{k}\) and \(7 \hat{i}-4 \hat{j}+7 \hat{k}\) is 11 cubic units, then value of $...

40Vector Algebra

If \(\overline{\mathrm{a}}=\mathrm{m} \overline{\mathrm{b}}+\mathrm{nc}\), where $$\overline{\mathrm{a}}=4 \hat{\mathrm{i}}+13 \hat{\mathrm{j}}-18 \hat{\mathrm{k}}, \overline{\mathrm{b}}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}...

41Vector Algebra

If \(\bar{a}\) and \(\bar{b}\) are two unit vectors such that \(\bar{a}+2 \bar{b}\) and \(5 \bar{a}-4 \bar{b}\) are perpendicular to each other, then the angle between \(\bar{a}\) and \(\bar{b}\) is

42Vector Algebra

If \(\bar{p}=\hat{i}+\hat{j}+\hat{k}\) and \(\bar{q}=\hat{i}-2 \hat{j}+\hat{k}\). Then a vector of magnitude \(5 \sqrt{3}\) units perpendicular to the vector \(\bar{q}\) and coplanar with \(\bar{p}\) and \(\bar{q}\) is

43Vector Algebra

If \((\bar{a} \times \bar{b}) \times \bar{c}=-5 \bar{a}+4 \bar{b}\) and \(\bar{a} \cdot \bar{b}=3\), then the value of \(\bar{a} \times(\bar{b} \times \bar{c})\) is

44Vector Algebra

The magnitude of the projection of the vector \(2 \hat{\mathbf{i}}+ 3\hat{\mathbf{j}}+\hat{\mathbf{k}}\) on the vector perpendicular to the plane containing the vectors \(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and $$\hat{\math...

45Vector Algebra

Let \(\bar{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\bar{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) be two vectors. If \(\bar{c}\) is a vector such that \(\bar{b} \times \bar{c}=\bar{b} \times \bar{a}\) a...

46Vector Algebra

If \(\bar{a}=\hat{\boldsymbol{i}}+\hat{\boldsymbol{j}}+\hat{\boldsymbol{k}}, \bar{b}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{j}}+\hat{\boldsymbol{k}}\) and \(\bar{c}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{j}}-\hat{\boldsymbol{k}}\) are three v...

47Vector Algebra

The polar co-ordinates of the point, whose Cartesian coordinates are \((-2 \sqrt{3}, 2)\), are

48Vector Algebra

If \(\bar{a}=\hat{\boldsymbol{i}}-\hat{\boldsymbol{k}}, \bar{b}=x \hat{\boldsymbol{i}}+\hat{\boldsymbol{j}}+(1-x) \hat{\boldsymbol{k}}\) and \(\bar{c}=y \hat{\boldsymbol{i}}+x \hat{\boldsymbol{j}}+(1+x-y) \hat{\boldsymbol{k}}\), then $$[\ba...