CBSE Class 12 Mathematics Vectors Algebra MCQs Set 08

Question. Find the sum of the vectors \(\vec{a} = \hat{i} - 2\hat{j} + \hat{k}\), \(\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}\) and \(\vec{c} = \hat{i} - 6\hat{j} - 7\hat{k}\).
(a) \(-4\hat{j} - \hat{k}\)
(b) \(-\hat{i} - 4\hat{j} - \hat{k}\)
(c) \(4\hat{j} + \hat{k}\)
(d) \(\hat{i} - 4\hat{j}\)
Answer: (a) \(-4\hat{j} - \hat{k}\)

Question. The magnitude of the vector \(\vec{a} = 3\hat{i} - 2\hat{j} + 6\hat{k}\) is equal to
(a) 6
(b) 7
(c) 7.5
(d) 8.5
Answer: (b) 7

Question. \((\vec{a} \cdot \hat{i})^2 + (\vec{a} \cdot \hat{j})^2 + (\vec{a} \cdot \hat{k})^2\) is equal to
(a) 1
(b) \(|\vec{a}|\)
(c) \(-\vec{a}\)
(d) \(|\vec{a}|^2\)
Answer: (d) \(|\vec{a}|^2\)

Question. If ABCD is a rhombus, whose diagonals intersect at E, then \(\vec{EA} + \vec{EB} + \vec{EC} + \vec{ED}\) equals
(a) \(\vec{0}\)
(b) \(\vec{AD}\)
(c) \(2\vec{BC}\)
(d) \(2\vec{AD}\)
Answer: (a) \(\vec{0}\)

Question. If \(\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}\) and \(\vec{b} = 4\hat{i} + 4\hat{j} - 2\hat{k}\) then find the angle between the vectors \(\vec{a}\) and \(\vec{b}\).
(a) 0
(b) \(\frac{\pi}{3}\)
(c) \(\frac{\pi}{2}\)
(d) \(\frac{\pi}{4}\)
Answer: (c) \(\frac{\pi}{2}\)

Question. The projection of the vector \(\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}\) on the vector \(\vec{b} = \hat{i} + 2\hat{j} + \hat{k}\) is
(a) \(\frac{10}{\sqrt{6}}\)
(b) \(\frac{10}{\sqrt{3}}\)
(c) \(\frac{5}{\sqrt{6}}\)
(d) \(\frac{5}{\sqrt{3}}\)
Answer: (a) \(\frac{10}{\sqrt{6}}\)

Question. If A and B are the points \((-3, 4, -8)\) and \((5, -6, 4)\) respectively, then find the ratio in which \(yz\)-plane divides AB.
(a) 5 : 2
(b) 7 : 5
(c) 3 : 5
(d) 5 : 3
Answer: (c) 3 : 5

Question. The vector in the direction of the vector \(\hat{i} - 2\hat{j} + 2\hat{k}\) that has magnitude 9 is
(a) \(\hat{i} - 2\hat{j} + 2\hat{k}\)
(b) \(\frac{\hat{i} - 2\hat{j} + 2\hat{k}}{3}\)
(c) \(3(\hat{i} - 2\hat{j} + 2\hat{k})\)
(d) \(9(\hat{i} - 2\hat{j} + 2\hat{k})\)
Answer: (c) \(3(\hat{i} - 2\hat{j} + 2\hat{k})\)

Question. If the angle between \(\hat{i} + \hat{k}\) and \(\hat{i} + \hat{j} + a\hat{k}\) is \(\frac{\pi}{3}\), then the value of a is
(a) 0 or 2
(b) \(-4\) or 0
(c) 0 or \(-2\)
(d) 2 or \(-2\)
Answer: (b) \(-4\) or 0

Question. If \(|\vec{a} - \vec{b}| = |\vec{a}| = |\vec{b}| = 1\), then the angle between \(\vec{a}\) and \(\vec{b}\) is
(a) \(\frac{\pi}{3}\)
(b) \(\frac{3\pi}{4}\)
(c) \(\frac{\pi}{2}\)
(d) \(\frac{\pi}{6}\)
Answer: (a) \(\frac{\pi}{3}\)

Question. If \(\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}\), \(\vec{b} = -\hat{i} + 2\hat{j} + \hat{k}\) and \(\vec{c} = 3\hat{i} + \hat{j} + 2\hat{k}\), then the value of \(\vec{a} \cdot (\vec{b} \times \vec{c})\) is
(a) \(-20\)
(b) \(-10\)
(c) 10
(d) 20
Answer: (b) \(-10\)

Question. The magnitude of each of the two vectors \(\vec{a}\) and \(\vec{b}\), having the same magnitude such that the angle between them is \(60^\circ\) and their scalar product is \(\frac{9}{2}\), is
(a) 2
(b) 3
(c) 4
(d) 5
Answer: (b) 3

Question. If \(\vec{u} = \hat{i} + 2\hat{j}\), \(\vec{v} = -2\hat{i} + \hat{j}\) and \(\vec{w} = 4\hat{i} + 3\hat{j}\), then find scalars \(x\) and \(y\) such that \(\vec{w} = x\vec{u} + y\vec{v}\).
(a) \(x = 4, y = -2\)
(b) \(x = 2, y = -1\)
(c) \(x = 3, y = 5\)
(d) \(x = -5, y = 2\)
Answer: (b) \(x = 2, y = -1\)

Question. Write the direction cosines of the vector \(-2\hat{i} + \hat{j} - 5\hat{k}\).
(a) \(\left(\frac{2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{5}{\sqrt{30}}\right)\)
(b) \(\left(\frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}}\right)\)
(c) \(\left(\frac{2}{\sqrt{30}}, \frac{-1}{\sqrt{30}}, \frac{-5}{\sqrt{30}}\right)\)
(d) None of the options
Answer: (b) \(\left(\frac{-2}{\sqrt{30}}, \frac{1}{\sqrt{30}}, \frac{-5}{\sqrt{30}}\right)\)

Question. Let \(\vec{a}\) and \(\vec{b}\) are non-collinear. If \(\vec{c} = (x-2)\vec{a} + \vec{b}\) and \(\vec{d} = (2x+1)\vec{a} - \vec{b}\) are collinear, then find the value of \(x\).
(a) \(\frac{2}{3}\)
(b) \(-\frac{1}{3}\)
(c) \(-\frac{2}{3}\)
(d) \(\frac{1}{3}\)
Answer: (d) \(\frac{1}{3}\)

Question. If \(\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}\) and \(\vec{b} = 3\hat{i} + 5\hat{j} - 2\hat{k}\), then \(|\vec{a} \times \vec{b}|\) is equal to
(a) \(\sqrt{507}\)
(b) \(\sqrt{506}\)
(c) \(\sqrt{508}\)
(d) \(\sqrt{509}\)
Answer: (a) \(\sqrt{507}\)

Question. \((\hat{i} + \hat{j}) \times (\hat{j} + \hat{k}) \cdot (\hat{k} + \hat{i})\) is equal to
(a) 0
(b) 1
(c) 2
(d) \(-1\)
Answer: (c) 2

Case Based MCQs

Case I : Read the following passage and answer the questions.
Ginni purchased an air plant holder which is in the shape of a tetrahedron.
Let A, B, C and D are the coordinates of the air plant holder where A \(\equiv (1, 1, 1)\), B \(\equiv (2, 1, 3)\), C \(\equiv (3, 2, 2)\) and D \(\equiv (3, 3, 4)\).

Question. Find the position vector of \(\vec{AB}\).
(a) \(-\hat{i} - 2\hat{k}\)
(b) \(2\hat{i} + \hat{k}\)
(c) \(\hat{i} + 2\hat{k}\)
(d) \(-2\hat{i} - \hat{k}\)
Answer: (c) \(\hat{i} + 2\hat{k}\)

Question. Find the position vector of \(\vec{AC}\).
(a) \(2\hat{i} - \hat{j} - \hat{k}\)
(b) \(2\hat{i} + \hat{j} + \hat{k}\)
(c) \(-2\hat{i} - \hat{j} + \hat{k}\)
(d) \(\hat{i} + 2\hat{j} + \hat{k}\)
Answer: (b) \(2\hat{i} + \hat{j} + \hat{k}\)

Question. Find the position vector of \(\vec{AD}\).
(a) \(2\hat{i} - 2\hat{j} - 3\hat{k}\)
(b) \(\hat{i} + \hat{j} - 3\hat{k}\)
(c) \(3\hat{i} + 2\hat{j} + 2\hat{k}\)
(d) \(2\hat{i} + 2\hat{j} + 3\hat{k}\)
Answer: (d) \(2\hat{i} + 2\hat{j} + 3\hat{k}\)

Question. Area of \(\Delta ABC =\)
(a) \(\frac{\sqrt{11}}{2}\) sq. units
(b) \(\frac{\sqrt{14}}{2}\) sq. units
(c) \(\frac{\sqrt{13}}{2}\) sq. units
(d) \(\frac{\sqrt{17}}{2}\) sq. units
Answer: (b) \(\frac{\sqrt{14}}{2}\) sq. units

Question. Find the unit vector along \(\vec{AD}\).
(a) \(\frac{1}{\sqrt{17}}(2\hat{i} + 2\hat{j} + 3\hat{k})\)
(b) \(\frac{1}{\sqrt{17}}(3\hat{i} + 3\hat{j} + 2\hat{k})\)
(c) \(\frac{1}{\sqrt{11}}(2\hat{i} + 2\hat{j} + 3\hat{k})\)
(d) \((2\hat{i} + 2\hat{j} + 3\hat{k})\)
Answer: (a) \(\frac{1}{\sqrt{17}}(2\hat{i} + 2\hat{j} + 3\hat{k})\)

Case II : Read the following passage and answer the questions.
Three slogans on chart papers are to be placed on a school bulletin board at the points A, B and C displaying A (Hub of Learning), B (Creating a better world for tomorrow) and C (Education comes first). The coordinates of points A, B and C are \((1, 4, 2)\), \((3, -3, -2)\) and \((-2, 2, 6)\) respectively.

Question. Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be the position vectors of points A, B and C respectively, then \(\vec{a} + \vec{b} + \vec{c}\) is equal to
(a) \(2\hat{i} + 3\hat{j} + 6\hat{k}\)
(b) \(2\hat{i} - 3\hat{j} - 6\hat{k}\)
(c) \(2\hat{i} + 8\hat{j} + 3\hat{k}\)
(d) \(2(7\hat{i} + 8\hat{j} + 3\hat{k})\)
Answer: (a) \(2\hat{i} + 3\hat{j} + 6\hat{k}\)

Question. Which of the following is not true?
(a) \(\vec{AB} + \vec{BC} + \vec{CA} = \vec{0}\)
(b) \(\vec{AB} + \vec{BC} - \vec{AC} = \vec{0}\)
(c) \(\vec{AB} + \vec{BC} - \vec{CA} = \vec{0}\)
(d) \(\vec{AB} - \vec{CB} + \vec{CA} = \vec{0}\)
Answer: (c) \(\vec{AB} + \vec{BC} - \vec{CA} = \vec{0}\)

Question. Area of \(\Delta ABC\) is
(a) 19 sq. units
(b) \(\sqrt{1937}\) sq. units
(c) \(\frac{1}{2}\sqrt{1937}\) sq. units
(d) \(\sqrt{1837}\) sq. units
Answer: (c) \(\frac{1}{2}\sqrt{1937}\) sq. units

Question. Suppose, if the given slogans are to be placed on a straight line, then the value of \(|\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}|\) will be equal to
(a) -1
(b) -2
(c) 2
(d) 0
Answer: (d) 0

Question. If \(\vec{a} = 2\hat{i} + 3\hat{j} + 6\hat{k}\), then unit vector in the direction of vector \(\vec{a}\) is
(a) \(\frac{2}{7}\hat{i} - \frac{3}{7}\hat{j} - \frac{6}{7}\hat{k}\)
(b) \(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k}\)
(c) \(\frac{3}{7}\hat{i} + \frac{2}{7}\hat{j} + \frac{6}{7}\hat{k}\)
(d) None of the options
Answer: (b) \(\frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k}\)

Assertion & Reasoning Based MCQs

Directions: In these questions, a statement of Assertion is followed by a statement of Reason. Choose the correct answer out of the following choices:
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: The magnitude of the resultant of vectors \(\vec{a} = 2\hat{i} + \hat{j} + \hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}\) is \(\sqrt{34}\).
Reason: The magnitude of a vector can never be negative.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Question. Assertion: The unit vector in the direction of sum of the vectors \(\hat{i} + \hat{j} + \hat{k}\), \(2\hat{i} - \hat{j} - \hat{k}\) and \(2\hat{j} + 6\hat{k}\) is \(-\frac{1}{7}(3\hat{i} + 2\hat{j} + 6\hat{k})\).
Reason: Let \(\vec{a}\) be a non-zero vector, then \(\frac{\vec{a}}{|\vec{a}|}\) is a unit vector parallel to \(\vec{a}\).
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (d) Assertion is wrong statement but Reason is correct statement.

Question. Assertion: Vectors \(\vec{a} = \hat{i} + \hat{j} - 3\hat{k}\) and \(\vec{b} = 2\hat{i} + \hat{j} + \hat{k}\) are perpendicular to each other.
Reason: \(\vec{a} \cdot \vec{b} = 0\)
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Question. Assertion: The adjacent sides of a parallelogram are along \(\vec{a} = \hat{i} + 2\hat{j}\) and \(\vec{b} = 2\hat{i} + \hat{j}\). The angle between the diagonals is \(150^\circ\).
Reason: Two vectors are perpendicular to each other if their dot product is zero.
(a) Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
(b) Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
(c) Assertion is correct statement but Reason is wrong statement.
(d) Assertion is wrong statement but Reason is correct statement.
Answer: (d) Assertion is wrong statement but Reason is correct statement.