Vector Algebra

The magnitude of vector \(\vec{a} = 3\hat{i} + 4\hat{j}\) is:

1. a5
2. b4
3. c3
4. d7

The vector \(\hat{i} \times \hat{j}\) equals:

1. a\(0\)
2. b\(-\hat{k}\)
3. c\(\hat{i}\)
4. d\(\hat{k}\)

If \(\vec{a} = 2\hat{i} - \hat{j}\) and \(\vec{b} = \hat{i} + 3\hat{j}\), then \(\vec{a} + \vec{b}\) is:

1. a\(\hat{i} + 2\hat{j}\)
2. b\(3\hat{i} + 2\hat{j}\)
3. c\(3\hat{i} - 2\hat{j}\)
4. d\(\hat{i} - 2\hat{j}\)

If \(\vec{a} \cdot \vec{b} = 0\) and both are non-zero, then the angle between them is:

1. a\(\dfrac{\pi}{2}\)
2. b\(\dfrac{\pi}{4}\)
3. c\(0\)
4. d\(\pi\)

The unit vector in direction of \(3\hat{i} + 4\hat{j}\) is:

1. a\(\hat{i} + \hat{j}\)
2. b\(3\hat{i} + 4\hat{j}\)
3. c\(\dfrac{3}{5}\hat{i} + \dfrac{4}{5}\hat{j}\)
4. d\(\dfrac{1}{7}(3\hat{i} + 4\hat{j})\)

If \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\) and \(\vec{b} = 2\hat{i} - \hat{j} + \hat{k}\), then \(\vec{a} \cdot \vec{b}\) is:

1. a1
2. b7
3. c5
4. d3

Cross product of \(\hat{i}\) and \(\hat{k}\) is:

1. a\(\hat{j}\)
2. b\(\hat{i}\)
3. c\(-\hat{j}\)
4. d\(0\)

Area of parallelogram with adjacent sides \(\vec{a} = \hat{i} + \hat{j}\) and \(\vec{b} = \hat{j} + \hat{k}\) is:

1. a\(\sqrt{3}\)
2. b\(\sqrt{2}\)
3. c\(1\)
4. d\(2\)

If vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are mutually perpendicular unit vectors, then \(|\vec{a} + \vec{b} + \vec{c}|\) is:

1. a\(3\)
2. b\(\sqrt{3}\)
3. c\(1\)
4. d\(\sqrt{2}\)

Position vector of midpoint of segment joining \(A(\vec{a})\) and \(B(\vec{b})\) is:

1. a\(\vec{a} - \vec{b}\)
2. b\(\dfrac{\vec{a} + \vec{b}}{2}\)
3. c\(\dfrac{\vec{a} - \vec{b}}{2}\)
4. d\(\vec{a} + \vec{b}\)

Assertion (A): The vectors \(2\hat{i} + 3\hat{j} - \hat{k}\) and \(-4\hat{i} - 6\hat{j} - 2\hat{k}\) are collinear.

Reason (R): Two vectors \(\vec{a}\) and \(\vec{b}\) are collinear if and only if \(\vec{b} = \lambda\vec{a}\) for some non-zero scalar \(\lambda\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): If the points A and B are \((1, 2, -3)\) and \((3, -4, 0)\), then \(|\overrightarrow{AB}| = 7\).

Reason (R): \(\overrightarrow{AB}\) is found by subtracting the position vector of B from the position vector of A.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): The direction cosines of the vector \(\vec{a} = \hat{i} + 2\hat{j} + 2\hat{k}\) are \(\dfrac{1}{3}, \dfrac{2}{3}, \dfrac{2}{3}\).

Reason (R): The sum of the squares of the direction cosines of any vector is \(1\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. Vector addition is commutative: \(\vec{a} + \vec{b} = \vec{b} + \vec{a}\).
2. II. Scalar multiplication distributes over vector addition: \(k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}\).

Which vector statements are correct?

1. aBoth
2. bII only
3. cI only
4. dNeither

1. I. The cross product is anticommutative: \(\vec{a} \times \vec{b} = -\vec{b} \times \vec{a}\).
2. II. The cross product is associative: \((\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})\).

Which statements are correct?

1. aBoth
2. bII only
3. cI only
4. dNeither

1. I. The projection of \(\vec{a}\) on \(\vec{b}\) is \(\dfrac{\vec{a} \cdot \vec{b}}{|\vec{a}|}\).
2. II. \(|\vec{a} \times \vec{b}|^{2} + (\vec{a} \cdot \vec{b})^{2} = |\vec{a}|^{2}|\vec{b}|^{2}\).

Which statements are correct?

1. aI only
2. bNeither
3. cBoth
4. dII only

Match each expression with what it represents (\(\theta\) is the angle between \(\vec{a}\) and \(\vec{b}\)):

1. a(i)-P, (ii)-Q, (iii)-R, (iv)-S
2. b(i)-Q, (ii)-P, (iii)-S, (iv)-R
3. c(i)-R, (ii)-S, (iii)-Q, (iv)-P
4. d(i)-S, (ii)-R, (iii)-P, (iv)-Q

For unit vectors \(\hat{i}\), \(\hat{j}\), \(\hat{k}\) along axes, match:

1. a(i)-Q, (ii)-P, (iii)-S, (iv)-R
2. b(i)-P, (ii)-Q, (iii)-R, (iv)-S
3. c(i)-R, (ii)-S, (iii)-Q, (iv)-P
4. d(i)-S, (ii)-R, (iii)-P, (iv)-Q

A drone flies in a straight line from its launch point P to its landing point Q. Relative to a fixed origin, their position vectors (in metres) are \(\vec{p} = 2\hat{i} - \hat{j} + 3\hat{k}\) and \(\vec{q} = 5\hat{i} + 5\hat{j} - 3\hat{k}\). The drone takes a photo at the point R that divides PQ internally in the ratio 2 : 1 (PR : RQ = 2 : 1). The position vector of R is:

1. a\(3\hat{i} + \hat{j} + \hat{k}\)
2. b\(8\hat{i} + 11\hat{j} - 9\hat{k}\)
3. c\(\dfrac{7}{2}\hat{i} + 2\hat{j}\)
4. d\(4\hat{i} + 3\hat{j} - \hat{k}\)

Find \(\lambda\) such that \(\vec{a} = \hat{i} + 2\hat{j} + \lambda\hat{k}\) and \(\vec{b} = 2\hat{i} - \hat{j} + 3\hat{k}\) are perpendicular.

1. a\(0\)
2. b\(1\)
3. c\(2\)
4. d\(-1\)