Mathematics ยท Physics

Vector Algebra and Calculus

192 Questions

Vector algebra involves mathematical operations on spatial quantities including dot products and cross products. These questions test the understanding of vector spaces and linear combinations. This topic is crucial for advanced mathematics and physics exams.

Dot and cross productsVector linear combinationsPerpendicular vector calculationsVector space dimensionsCollinear points and vectors

Vector Algebra and Calculus Questions

Multiple choice

Given two vectors (\vec{A} = 2\hat{i} - 3\hat{j}) and (\vec{B} = 4\hat{i} + 5\hat{j}), find the vector (\vec{A} + \vec{B}).

  1. 6\hat{i} + 2\hat{j}

  2. 6\hat{i} - 8\hat{j}

  3. 8\hat{i} + 2\hat{j}

  4. 8\hat{i} - 8\hat{j}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Vector addition is performed component-wise. Therefore, (\vec{A} + \vec{B} = (2\hat{i} - 3\hat{j}) + (4\hat{i} + 5\hat{j}) = (2 + 4)\hat{i} + (-3 + 5)\hat{j} = 6\hat{i} + 2\hat{j}).

Multiple choice

If (\vec{A} = \langle 2, 3, 4 \rangle) and (\vec{B} = \langle -1, 5, -2 \rangle), find the vector (\vec{A} - \vec{B}).

  1. \langle 3, -2, 6 \rangle\

  2. \langle 3, 8, 2 \rangle\

  3. \langle -3, 8, 6 \rangle\

  4. \langle -3, -2, 2 \rangle\
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Vector subtraction is also performed component-wise. Therefore, (\vec{A} - \vec{B} = \langle 2, 3, 4 \rangle\ - \langle -1, 5, -2 \rangle\ = \langle 2 - (-1), 3 - 5, 4 - (-2) \rangle\ = \langle 3, -2, 6 \rangle).

Multiple choice

Given a vector (\vec{A} = 5\hat{i} - 2\hat{j}), find the vector (2\vec{A}).

  1. 10\hat{i} - 4\hat{j}

  2. 10\hat{i} + 4\hat{j}

  3. 20\hat{i} - 4\hat{j}

  4. 20\hat{i} + 4\hat{j}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Scalar multiplication is performed by multiplying each component of the vector by the scalar. Therefore, (2\vec{A} = 2(5\hat{i} - 2\hat{j}) = 10\hat{i} - 4\hat{j}).

Multiple choice

What is the dot product of the vectors (\vec{A} = \langle 1, 2, 3 \rangle) and (\vec{B} = \langle 4, -5, 6 \rangle)?

  1. 3

  2. 12

  3. 21

  4. 30
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The dot product of two vectors is given by the formula (\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z). Substituting the values of the components, we get (\vec{A} \cdot \vec{B} = (1)(4) + (2)(-5) + (3)(6) = 4 - 10 + 18 = 12).

Multiple choice

Find the cross product of the vectors (\vec{A} = \hat{i} + 2\hat{j} - 3\hat{k}) and (\vec{B} = 2\hat{i} - \hat{j} + 4\hat{k}).

  1. \langle -11, -2, 5 \rangle\

  2. \langle -11, 2, 5 \rangle\

  3. \langle 11, -2, -5 \rangle\

  4. \langle 11, 2, -5 \rangle\
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The cross product of two vectors is given by the formula (\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix}). Expanding the determinant, we get (\vec{A} \times \vec{B} = \langle (2)(-4) - (-1)(8), (-1)(2) - (1)(4), (1)(-1) - (2)(2) \rangle\ = \langle -11, 2, 5 \rangle).

Multiple choice

Which of the following vectors is perpendicular to both (\vec{A} = \hat{i} + 2\hat{j} - 3\hat{k}) and (\vec{B} = 2\hat{i} - \hat{j} + 4\hat{k})?

  1. \langle 1, 1, 1 \rangle\

  2. \langle 1, -1, 1 \rangle\

  3. \langle -1, 1, -1 \rangle\

  4. \langle -1, -1, 1 \rangle\
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A vector is perpendicular to two other vectors if its dot product with both of them is zero. Let (\vec{C} = \langle x, y, z \rangle) be the vector we are looking for. Then, (\vec{A} \cdot \vec{C} = (1)(x) + (2)(y) + (-3)(z) = 0) and (\vec{B} \cdot \vec{C} = (2)(x) + (-1)(y) + (4)(z) = 0). Solving these two equations simultaneously, we get (\vec{C} = \langle -1, 1, -1 \rangle).

Multiple choice

Find the angle between the vectors (\vec{A} = \hat{i} + \hat{j}) and (\vec{B} = \hat{i} - \hat{j}).

  1. 30\degree

  2. 45\degree

  3. 60\degree

  4. 90\degree
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The angle between two vectors is given by the formula (\theta = \cos^{-1}\left(\frac{\vec{A} \cdot \vec{B}}{|\vec{A}||\vec{B}|}\right)). Substituting the values of the vectors, we get (\theta = \cos^{-1}\left(\frac{(1)(1) + (1)(-1)}{\sqrt{1^2 + 1^2}\sqrt{1^2 + (-1)^2}}\right) = \cos^{-1}\left(\frac{0}{\sqrt{2}\sqrt{2}}\right) = \cos^{-1}(0) = 90\degree).

Multiple choice

Given a vector (\vec{A} = 3\hat{i} - 4\hat{j}), find the unit vector in the same direction as (\vec{A}).

  1. \frac{3}{5}\hat{i} - \frac{4}{5}\hat{j}

  2. \frac{3}{7}\hat{i} - \frac{4}{7}\hat{j}

  3. \frac{4}{5}\hat{i} + \frac{3}{5}\hat{j}

  4. \frac{4}{7}\hat{i} + \frac{3}{7}\hat{j}
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The unit vector in the same direction as (\vec{A}) is given by (\hat{A} = \frac{\vec{A}}{|\vec{A}|}). Substituting the values of (\vec{A}) and (|\vec{A}|), we get (\hat{A} = \frac{3\hat{i} - 4\hat{j}}{\sqrt{3^2 + (-4)^2}} = \frac{3\hat{i} - 4\hat{j}}{5} = \frac{3}{5}\hat{i} - \frac{4}{5}\hat{j}).

Multiple choice

Which of the following vectors is parallel to the vector (\vec{A} = 2\hat{i} - 3\hat{j})?

  1. \hat{i} - \frac{3}{2}\hat{j}

  2. 2\hat{i} + 3\hat{j}

  3. 3\hat{i} - 2\hat{j}

  4. 6\hat{i} - 9\hat{j}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Two vectors are parallel if they have the same direction. This means that one vector can be obtained by multiplying the other vector by a scalar. In this case, we can see that (3\hat{i} - 2\hat{j} = \frac{3}{2}(2\hat{i} - 3\hat{j})). Therefore, (\vec{A}) and (3\hat{i} - 2\hat{j}) are parallel.

Multiple choice

Which of the following vectors is a linear combination of the vectors (\vec{A} = \hat{i} + 2\hat{j}) and (\vec{B} = 2\hat{i} - \hat{j})?

  1. \hat{i} + \hat{j}

  2. 2\hat{i} + \hat{j}

  3. 3\hat{i} - \hat{j}

  4. 4\hat{i} + 2\hat{j}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A vector (\vec{C}) is a linear combination of the vectors (\vec{A}) and (\vec{B}) if there exist scalars (a) and (b) such that (\vec{C} = a\vec{A} + b\vec{B}). In this case, we can see that (3\hat{i} - \hat{j} = 3(\hat{i} + 2\hat{j}) + (-1)(2\hat{i} - \hat{j})). Therefore, (\vec{C} = 3\hat{i} - \hat{j}) is a linear combination of (\vec{A}) and (\vec{B}).

Multiple choice

Find the projection of the vector (\vec{A} = 2\hat{i} + 3\hat{j}) onto the vector (\vec{B} = 4\hat{i} - \hat{j}).

  1. \frac{10}{17}\hat{i} - \frac{3}{17}\hat{j}

  2. \frac{14}{17}\hat{i} + \frac{1}{17}\hat{j}

  3. \frac{14}{17}\hat{i} - \frac{1}{17}\hat{j}

  4. \frac{10}{17}\hat{i} + \frac{3}{17}\hat{j}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The projection of a vector (\vec{A}) onto a vector (\vec{B}) is given by the formula (\text{proj}{\vec{B}}\vec{A} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|^2}\vec{B}). Substituting the values of the vectors, we get (\text{proj}{\vec{B}}\vec{A} = \frac{(2)(4) + (3)(-1)}{(4^2 + (-1)^2)}(4\hat{i} - \hat{j}) = \frac{8 - 3}{17}(4\hat{i} - \hat{j}) = \frac{5}{17}(4\hat{i} - \hat{j}) = \frac{20}{17}\hat{i} - \frac{5}{17}\hat{j}).

Multiple choice

Which of the following vectors is orthogonal to both (\vec{A} = \hat{i} + 2\hat{j} - 3\hat{k}) and (\vec{B} = 2\hat{i} - \hat{j} + 4\hat{k})?

  1. \hat{i} + \hat{j} + \hat{k}

  2. \hat{i} - \hat{j} - \hat{k}

  3. \hat{i} + \hat{j} - \hat{k}

  4. \hat{i} - \hat{j} + \hat{k}
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A vector is orthogonal to two other vectors if its dot product with both of them is zero. Let (\vec{C} = \langle x, y, z \rangle) be the vector we are looking for. Then, (\vec{A} \cdot \vec{C} = (1)(x) + (2)(y) + (-3)(z) = 0) and (\vec{B} \cdot \vec{C} = (2)(x) + (-1)(y) + (4)(z) = 0). Solving these two equations simultaneously, we get (\vec{C} = \langle 1, 1, -1 \rangle).