Vector Algebra and Components Practice Quiz (9 questions)

1. A measurement of a system's temperature is specified by one numerical value and has no direction. How is this quantity classified?

A. A direction

B. A vector

C. A force

D. A scalar

Answer: (D)

Explanation

A scalar is defined by a single value and has magnitude but no direction. A vector has direction, a force is a vector quantity, and direction is not a quantity classification.

2. A model represents the acceleration of a vehicle with an arrow whose length shows its size and whose orientation shows where it is headed. Which type of quantity is being represented?

A. A magnitude

B. A vector

C. A scalar

D. A force

Answer: (B)

Explanation

Acceleration has magnitude and direction, so it is a vector represented by an arrow. A scalar has magnitude without direction, magnitude is only one property, and force is a different vector quantity.

3. A two-dimensional vector has an x-coordinate of $-4$ and a y-coordinate of $9$. Which ordered pair gives its Cartesian coordinates?

A. $(9,-4)$

B. $(4,9)$

C. $(-4,-9)$

D. $(-4,9)$

Answer: (D)

Explanation

Cartesian coordinates express a two-dimensional vector as the ordered pair $A=(x,y)$, so the x-coordinate comes first and the y-coordinate second. Reversing the pair swaps the axes, while changing either sign changes the specified coordinate.

4. Four vectors are assigned the coordinate pairs $P=(3,6)$, $Q=(-6,6)$, $R=(-3,-2)$, and $S=(11,-5)$. Which statement correctly identifies the sign combinations represented?

A. The pairs represent negative-positive twice, positive-positive once, and negative-negative once.

B. The pairs represent all four combinations: positive-positive, negative-positive, negative-negative, and positive-negative.

C. The pairs represent positive-negative twice, negative-positive once, and positive-positive once.

D. The pairs represent positive-positive twice, negative-negative once, and positive-negative once.

Answer: (B)

Explanation

The signs of the ordered x- and y-coordinates in the four pairs cover all four possible combinations. The alternative groupings misread one or more pairs, such as treating $(-6,6)$ as positive-positive or $(11,-5)$ as negative-positive.

5. What does a vector component represent in trigonometric resolution?

A. A projection of the vector onto one of the coordinate axes

B. The angle measured between the vector and the positive x-axis

C. The point where the vector’s tail is placed in the coordinate plane

D. The vector’s magnitude calculated from its perpendicular projections

Answer: (A)

Explanation

A vector component is the projection of the vector onto the x- or y-axis, and the vector is the sum of those components. The angle describes the vector’s orientation rather than a projection onto an axis.

6. What is the magnitude of the vector with components $A_x=13$ and $A_y=7$?

A. Approximately 13.76

B. Approximately 20.00

C. Approximately 6.00

D. Approximately 14.76

Answer: (D)

Explanation

The perpendicular components give a magnitude of $\sqrt{13^2+7^2}=\sqrt{218}\approx14.76$. Adding the components directly gives 20, but that does not account for their perpendicular directions.

7. What is the function of a unit vector when expressing a vector in component notation?

A. It gives the total magnitude of the vector while its coefficients specify the coordinate axes.

B. It converts the vector’s Cartesian coordinates into an angle and a scalar magnitude.

C. It identifies the vector’s point of application while its coefficients give its orientation.

D. It indicates the direction of each component while its coefficient gives that component’s magnitude.

Answer: (D)

Explanation

A unit vector provides the direction associated with an axis, and its component coefficient scales it to represent the vector’s contribution along that axis. The total magnitude instead requires combining the components, rather than reading it directly from the unit vectors.

8. When does a vector have a point of application?

A. When it represents an acceleration

B. When it represents a velocity

C. When it represents a force

D. When it represents a displacement

Answer: (C)

Explanation

A point of application is a property of a vector when that vector represents a force. Displacement, velocity, and acceleration are vectors, but the material does not assign them a point of application.

9. How do the x- and y-coordinates of a Cartesian vector determine its quadrant?

A. Their numerical magnitudes determine the vector's line of action.

B. Their order determines whether the vector has a point of application.

C. Their sum determines the vector's direction and sense.

D. Their positive or negative signs identify the quadrant.

Answer: (D)

Explanation

The signs of the two coordinates indicate whether the vector lies to the right or left of the y-axis and above or below the x-axis, identifying its quadrant. The coordinate magnitudes describe its component values, but they do not by themselves identify the quadrant.