Question 1: Palette 1/10

Resnick Haliday MCQ

Concept: Scalar quantities are physical quantities that have magnitude only and no direction. Because they do not involve directional orientation, scalar quantities obey the standard rules of ordinary algebra and arithmetic for operations such as addition and subtraction.

Which one of the following statements is true concerning scalar quantities?

A

Scalar quantities must be represented by base units.

B

Scalar quantities have both magnitude and direction.

C

Scalar quantities can be added to vector quantities using rules of trigonometry.

D

Scalar quantities can be added to other scalar quantities using rules of ordinary addition.

Question 2: Palette 2/10

Resnick Haliday MCQ

Concept: A vector quantity is a physical quantity that possesses both magnitude and direction, and obeys the laws of vector addition. In contrast, scalar quantities have magnitude only and lack any directional component.

Which one of the following quantities is a vector quantity?

A

the age of the Earth

B

the mass of a freight train

C

the Earth's pull on your body

D

the temperature of a hot cup of coffee

Question 3: Palette 3/10

Resnick Haliday MCQ

Concept: Vectors have both magnitude and direction, whereas scalars have only magnitude. A vector quantity can be resolved into components along chosen coordinate axes, and these components can be either positive, zero, or negative. A vector whose magnitude is zero (a null vector) has all its components equal to zero.

Which one of the following statements concerning vectors and scalars is false?

A

In calculations, the vector components of a vector may be used in place of the vector itself.

B

It is possible to use vector components that are not perpendicular.

C

A scalar component may be either positive or negative.

D

A vector that is zero may have components other than zero.

Question 4: Palette 4/10

Resnick Haliday MCQ

Concept: For a set of $n$ coplanar forces of equal magnitude to be in equilibrium, they must be symmetrically distributed around a point so that their vector sum equals zero. The total angle surrounding a point is $360^\circ$ (or $2\pi\text{ radians}$). Therefore, the angle between any two adjacent forces in a symmetric distribution is obtained by dividing $360^\circ$ by the number of forces $n$.

Twelve coplanar forces (all of equal magnitude) maintain a body in equilibrium, then the angle between any two adjacent forces is

A

$15^\circ$

B

$30^\circ$

C

$45^\circ$

D

$60^\circ$

Question 5: Palette 5/10

Resnick Haliday MCQ

Concept: In vector addition for a triangle formed by vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$, the vectors representing the sides satisfy $\vec{A} + \vec{B} + \vec{C} = 0$ or $\vec{C} = \vec{A} - \vec{B}$ (or $\vec{C} = \vec{A} + \vec{B}$), depending on the orientation of the vector sides. Therefore, the third side can be represented by either $\vec{C} = \vec{A} + \vec{B}$ or $\vec{C} = \vec{A} - \vec{B}$ (or their negatives, which have the same length).

If vectors $\vec{A} = \hat{i} + 2\hat{j} + 4\hat{k}$ and $\vec{B} = 5\hat{i}$ represent the two sides of a triangle, then the third side of the triangle can have length equal to

A

$6$

B

$\sqrt{56}$

C

both (a) and (b)

D

none of the above

Question 6: Palette 6/10

Resnick Haliday MCQ

Concept: For any two vectors $\vec{a}$ and $\vec{b}$, the triangle inequality of vector addition states that the magnitude of the sum of two vectors is always less than or equal to the sum of their individual magnitudes, i.e., $\vert{}\vec{a} + \vec{b}\vert{} \le \vert{}\vec{a}\vert{} + \vert{}\vec{b}\vert{}$. Equality holds when the two vectors are collinear and point in the same direction.

Mark the correct statement.

A

$\vert{}\vec{a} + \vec{b}\vert{} \ge \vert{}\vec{a}\vert{} + \vert{}\vec{b}\vert{}$

B

$\vert{}\vec{a} + \vec{b}\vert{} \le \vert{}\vec{a}\vert{} + \vert{}\vec{b}\vert{}$

C

$\vert{}\vec{a} - \vec{b}\vert{} \ge \vert{}\vec{a}\vert{} + \vert{}\vec{b}\vert{}$

D

All of the above

Question 7: Palette 7/10

Resnick Haliday MCQ

Concept: In vector addition, the components of a resultant vector are the sums of the corresponding components of the individual vectors ($C_x = A_x + B_x$ and $C_y = A_y + B_y$). By comparing these component equations with the given expressions, we can determine the directional angles and alignments of vectors $\vec{A}$ and $\vec{B}$ relative to the coordinate axes.

Two vectors $\vec{A}$ and $\vec{B}$ are added together to form the vector $\vec{C} = \vec{A} + \vec{B}$. The relationship between the components of these vectors is given by $C_x = A\cos 30^\circ + B$ and $C_y = -A\sin 30^\circ$. Which statement best describes the orientation of these vectors?

A

Vector $\vec{A}$ makes an angle of $30^\circ$ below the positive x-axis, and vector $\vec{B}$ points along the positive x-axis.

B

Vector $\vec{A}$ points in the negative x-direction, while vector $\vec{B}$ points in the positive y-direction.

C

Vector $\vec{A}$ points in the negative y-direction, while vector $\vec{B}$ points in the positive x-direction.

D

Vector $\vec{A}$ makes an angle of $30^\circ$ above the positive x-axis, and vector $\vec{B}$ points along the negative x-axis.

Question 8: Palette 8/10

Resnick Haliday MCQ

Concept: When the vector sum of three vectors is zero ($\vec{A} + \vec{B} + \vec{C} = 0$), any one vector is equal to the negative of the sum of the other two, i.e., $\vec{B} = -(\vec{A} + \vec{C})$. If two vectors are antiparallel (pointing in opposite directions), their vector sum is along the direction of the larger vector, with a magnitude equal to the difference of their magnitudes.

Three vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ add together to yield zero: $\vec{A} + \vec{B} + \vec{C} = 0$. The vectors $\vec{A}$ and $\vec{C}$ point in opposite directions and their magnitudes are related by the expression: $A = 2C$. Which one of the following conclusions is correct?

A

$\vec{A}$ and $\vec{B}$ have equal magnitudes and point in opposite directions.

B

$\vec{B}$ and $\vec{C}$ have equal magnitudes and point in the same direction.

C

$\vec{B}$ and $\vec{C}$ have equal magnitudes and point in opposite directions.

D

$\vec{A}$ and $\vec{B}$ point in the same direction, but $A$ has twice the magnitude of $B$.

Question 9: Palette 9/10

Resnick Haliday MCQ

Concept: System of linear vector equations and solving for unknown vectors by elimination.

Given that $\vec{A} + 2\vec{B} = x_1\hat{i} + y_1\hat{j}$ and $2\vec{A} - \vec{B} = x_2\hat{i} + y_2\hat{j}$, what is $\vec{A}$?

A

$\vec{A} = \frac{1}{5}(x_1 + 2x_2)\hat{i} + \frac{1}{5}(y_1 + 2y_2)\hat{j}$

B

$\vec{A} = \frac{1}{5}(x_1 + 2x_2)\hat{i} - \frac{1}{5}(y_1 + 2y_2)\hat{j}$

C

$\vec{A} = \frac{1}{5}(x_1 + 4x_2)\hat{i} + \frac{1}{5}(y_1 + 2y_2)\hat{j}$

D

$\vec{A} = \frac{1}{5}(x_1 + 4x_2)\hat{i} + \frac{1}{5}(y_1 + 4y_2)\hat{j}$

Question 10: Palette 10/10

Resnick Haliday MCQ

Concept: Rotation of coordinate axes and transformation of vector components under rotation.

The vector $\vec{A}$ has components $+5$ and $+7$ along the $x$-axes and $y$-axes, respectively. Along a set of axes rotated $90^\circ$ degrees counterclockwise relative to the original axes, the vector's components are

A

$-7; -5$

B

$7; -5$

C

$-7; 5$

D

$7; 5$