AP Precalculus Quiz: Vector Valued Functions

What this quiz covers

This quiz focuses on Vector Valued Functions, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The position vector of the projectile at $t=2$ seconds. (correct answer)
2. The acceleration vector, which is constant for all $t$.
3. The launch angle $a$ measured from the horizontal axis.
4. The total distance traveled from $t=0$ to $t=2$.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, r(t) represents the position vector of a projectile at time t, and r(2) means substituting t = 2 into the function. Choice A is correct because r(2) = ⟨(v cos a)·2, (v sin a)·2 - 4.9·2²⟩ gives the exact position coordinates at 2 seconds. Choice B is incorrect because acceleration is the second derivative, not the position function itself. To help students: Connect vector notation to physical meaning, practice evaluating functions at specific times, and reinforce that r(t) tracks position over time. Watch for: confusing position, velocity, and acceleration vectors, or misunderstanding function evaluation.

Question 2

1. $\langle 6,5\rangle$ (correct answer)
2. $\langle 3,10\rangle$
3. $\langle 5,6\rangle$
4. $\langle 1.5,5\rangle$

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and matrix transformations for scaling operations. Scaling matrices multiply specific components by different factors, useful in robotics for adjusting path dimensions independently. In this problem, the scaling matrix S = [[2,0],[0,1]] scales the x-component by 2 and leaves the y-component unchanged. Choice A is correct because S⟨3, 5⟩ = [[2,0],[0,1]] × [3,5] = [2×3 + 0×5, 0×3 + 1×5] = [6, 5] = ⟨6, 5⟩. Choice B is incorrect because it appears to double the y-component instead of the x-component. To help students: Understand diagonal matrices as independent scalers, practice matrix-vector multiplication, and visualize geometric effects of scaling. Watch for: confusing which component gets scaled by which diagonal entry, arithmetic errors in multiplication.

Question 3

1. $\langle 4,-1\rangle$
2. $\langle -1,4\rangle$ (correct answer)
3. $\langle -4,1\rangle$
4. $\langle 1,4\rangle$

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and matrix transformations in engineering applications. Matrix-vector multiplication is fundamental in transforming vectors, with rotation matrices being a key application in camera systems and robotics. In this problem, we multiply the rotation matrix A = [[0,-1],[1,0]] by vector p = ⟨4, 1⟩. Choice B is correct because Ap = [[0,-1],[1,0]] × [4,1] = [0×4 + (-1)×1, 1×4 + 0×1] = [-1, 4] = ⟨-1, 4⟩, which represents a 90° counterclockwise rotation. Choice A is incorrect because it appears to be the original vector with components swapped, not the result of the matrix multiplication. To help students: Practice matrix-vector multiplication step by step, visualize geometric transformations, and verify results by checking rotation properties. Watch for: errors in matrix multiplication order, sign errors in calculations.

Question 4

1. The position vector at $t=2$ (correct answer)
2. The speed at $t=2$
3. The direction angle of flight
4. A 2D point with a nonzero $k$-value

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and evaluating position vectors at specific parameter values. Vector-valued functions r(t) give position as a function of time, and evaluating at a specific t-value gives the position at that instant. In this problem, r(t) = ⟨3t, 2t², 0⟩ represents a drone's path, and we need to interpret r(2). Choice A is correct because r(2) = ⟨3(2), 2(2)², 0⟩ = ⟨6, 8, 0⟩ represents the position vector (location) of the drone at time t = 2. Choice B is incorrect because r(2) gives position, not speed; speed would require calculating |r'(2)|. To help students: Distinguish between position, velocity, and speed in vector contexts, practice substituting parameter values, and interpret results physically. Watch for: confusion between position and velocity vectors, misunderstanding what vector function evaluation represents.

Question 5

1. Decrease $a$ while keeping $v$ fixed.
2. Increase $a$ while keeping $v$ fixed. (correct answer)
3. Decrease $t$ while keeping $v$ and $a$ fixed.
4. Replace $-4.9t^2$ with $-4.9t$ to remove curvature.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, the initial vertical velocity component is (v sin a), which depends on both v and angle a. Choice B is correct because increasing a (while keeping v fixed) increases sin a for angles between 0° and 90°, thus increasing the vertical velocity component. Choice A is incorrect because decreasing a would reduce sin a and lower the initial vertical velocity. To help students: Use unit circle knowledge to understand how sin a changes with angle, practice decomposing velocity vectors, and connect mathematical expressions to physical motion. Watch for: confusion about trigonometric function behavior, or mixing up horizontal and vertical components.

Question 6

1. It rotates the path by angle $a$ without changing range.
2. It makes both components larger in magnitude for each $t$. (correct answer)
3. It changes only the vertical component because gravity acts downward.
4. It adds a constant $k$-component to the position vector.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, the function r(t) = ⟨(v cos a)t, (v sin a)t - 4.9t²⟩ represents projectile motion, where v is the initial velocity magnitude. Choice B is correct because increasing v scales both velocity components proportionally - the horizontal component (v cos a)t and vertical component (v sin a)t both increase linearly with v, making the overall path larger. Choice A is incorrect because changing v doesn't rotate the path; the angle a controls direction. To help students: Emphasize how parameters multiply through expressions, use physical examples like throwing a ball harder, and practice identifying which parameters affect which components. Watch for: confusion between velocity magnitude v and angle a, or thinking gravity term -4.9t² depends on v.

Question 7

1. A 90° rotation about the origin.
2. A reflection across the $x$-axis. (correct answer)
3. A uniform scaling by factor 2 in both directions.
4. A translation right by 1 unit and up by 1 unit.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, matrix A = [[1,0],[0,-1]] is applied to vector r(t), transforming its components. Choice B is correct because this matrix keeps x-components unchanged (multiplied by 1) while negating y-components (multiplied by -1), creating a reflection across the x-axis. Choice A is incorrect because a 90° rotation requires a different matrix structure with sine and cosine terms. To help students: Practice matrix-vector multiplication step by step, visualize transformations geometrically, and memorize common transformation matrices. Watch for: confusing reflection matrices with rotation matrices, or misapplying matrix multiplication rules.

Question 8

1. The direction of motion in the $i$-$j$ plane (correct answer)
2. The segment length $L$
3. The time variable $t$ becomes constant
4. The motion gains a $k$-component automatically

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and rotation matrices in engineering applications. Vector-valued functions can incorporate transformation matrices like R(a) that rotate vectors, commonly used in robotics and mechanical systems. In this problem, r(t) = R(a)⟨Lt, 0, 0⟩ represents a robot arm where R(a) is a rotation matrix and L is segment length. Choice A is correct because R(a) is a rotation matrix that changes the direction of the vector in the i-j plane, rotating the arm's orientation by angle a. Choice B is incorrect because L appears as a scalar multiplier inside the vector, not affected by the rotation matrix R(a). To help students: Practice matrix-vector multiplication, visualize rotations geometrically, and connect mathematical transformations to physical movements. Watch for: confusion about what rotation matrices do, mixing up scalar parameters with transformation effects.

Question 9

1. $\langle v\cos a,\,v\sin a-9.8t\rangle$ (correct answer)
2. $\langle (v\cos a)t,\,(v\sin a)t-4.9t^2\rangle$
3. $\langle v\sin a,\,v\cos a-9.8t\rangle$
4. $\langle 0,\,-9.8\rangle$

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, velocity v(t) is the derivative of position r(t) = ⟨(v cos a)t, (v sin a)t - 4.9t²⟩. Choice A is correct because differentiating gives v(t) = ⟨v cos a, v sin a - 9.8t⟩, where the constant terms remain and the t² term becomes linear. Choice B is incorrect as it shows the position function, not its derivative. To help students: Practice differentiation of vector functions component-wise, understand physical meaning of derivatives (position → velocity → acceleration), and recognize standard projectile motion patterns. Watch for: forgetting to differentiate, confusing position with velocity, or errors in power rule application.

Question 10

1. It doubles the $y$-values but leaves $x$ unchanged.
2. It doubles the $x$-values but leaves $y$ unchanged. (correct answer)
3. It rotates the path 180° about the origin.
4. It swaps the $x$- and $y$-components of the vector.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, matrix S = [[2,0],[0,1]] is a diagonal scaling matrix applied to r(t). Choice B is correct because this matrix multiplies x-components by 2 (stretching horizontally) while multiplying y-components by 1 (leaving them unchanged). Choice A is incorrect because it reverses which component is scaled. To help students: Work through matrix multiplication explicitly showing S·⟨x,y⟩ = ⟨2x,y⟩, visualize diagonal matrices as independent scalings, and practice identifying transformation effects. Watch for: mixing up which diagonal entry affects which component, or misunderstanding scaling transformations.

Question 11

1. Increase $v$ while keeping $a$ the same. (correct answer)
2. Decrease $v$ while keeping $a$ the same.
3. Set $a=0$ so the projectile has maximum airtime.
4. Apply $A=\begin{pmatrix}1&0\\0&-1\end{pmatrix}$ to reverse gravity.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, horizontal range depends on both initial velocity v and launch angle a, with the range formula being proportional to v² sin(2a)/g. Choice A is correct because increasing v while keeping a constant increases the range quadratically - doubling v quadruples the range. Choice C is incorrect because setting a = 0 means horizontal launch with zero range (projectile immediately hits ground). To help students: Derive or understand the range formula, recognize that both speed and angle matter for range, and know that maximum range occurs at 45° for level ground. Watch for: thinking only angle matters for range, or misunderstanding that a = 0 means horizontal launch, not maximum range.

Question 12

1. It reflects the path across the $y$-axis.
2. It scales both components by 0.5.
3. It rotates vectors 90° counterclockwise about the origin. (correct answer)
4. It adds $\langle 1,1\rangle$ to every position vector.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, matrix R = [[0,-1],[1,0]] is the standard 90° counterclockwise rotation matrix. Choice C is correct because R transforms ⟨x,y⟩ to ⟨-y,x⟩, which geometrically rotates vectors 90° counterclockwise about the origin. Choice A is incorrect because reflection across y-axis would use matrix [[-1,0],[0,1]]. To help students: Memorize the standard rotation matrix, verify transformations by testing simple vectors like ⟨1,0⟩, and understand how matrix entries relate to geometric transformations. Watch for: confusing rotation direction (clockwise vs counterclockwise), or mixing up rotation and reflection matrices.

Question 13

1. Increase $a$ while keeping $v$ fixed. (correct answer)
2. Decrease $a$ while keeping $v$ fixed.
3. Increase $t$ while keeping $v$ and $a$ fixed.
4. Multiply the entire vector by $-1$ to steepen it.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, the initial trajectory steepness depends on the ratio of vertical to horizontal velocity components: (v sin a)/(v cos a) = tan a. Choice A is correct because increasing angle a increases tan a for angles between 0° and 90°, making the initial path steeper. Choice B is incorrect because decreasing a would reduce tan a and make the path less steep. To help students: Connect steepness to slope concepts, use trigonometric ratios to analyze trajectories, and visualize how launch angle affects projectile paths. Watch for: confusion about what 'steeper' means mathematically, or misunderstanding angle effects on trajectory.

Question 14

1. The $(v\cos a)t$ term in the $x$-component.
2. The $(v\sin a)t$ term in the $y$-component.
3. The $-4.9t^2$ term in the $y$-component. (correct answer)
4. The angle $a$ because it always forces a parabolic path.

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter effects. Vector-valued functions represent dynamic systems with vectors dependent on parameters, describing motion or transformations in space. In this problem, r(t) = ⟨(v cos a)t, (v sin a)t - 4.9t²⟩ models projectile motion under gravity. Choice C is correct because the -4.9t² term in the y-component represents gravitational acceleration (half of -9.8 m/s²), causing downward curvature as time increases. Choice D is incorrect because angle a affects initial direction but doesn't cause curvature - only the quadratic term does. To help students: Connect mathematical terms to physical phenomena, recognize that linear terms create straight motion while quadratic terms create curvature, and understand gravity's mathematical representation. Watch for: thinking all terms contribute to curvature, or not recognizing the physical significance of -4.9 as half of gravitational acceleration.

Question 15

1. The current vector $\langle 2,1,0\rangle$
2. The ship’s heading direction relative to $i$ (correct answer)
3. The magnitude of the current only
4. The $k$-component of the ship’s speed

Explanation: This question tests AP Precalculus skills: understanding vector-valued functions and parameter interpretation in navigation contexts. Vector-valued functions can model velocity as the sum of multiple vector components, such as a ship's engine velocity plus ocean current. In this problem, v = ⟨s cos a, s sin a, 0⟩ + ⟨2, 1, 0⟩ represents ground velocity as the sum of ship's heading velocity and current velocity. Choice B is correct because the parameter a appears in the trigonometric expressions (cos a, sin a) which determine the direction of the ship's heading vector relative to the positive i-axis. Choice A is incorrect because ⟨2, 1, 0⟩ is the current vector which is independent of parameter a. To help students: Decompose vector sums into components, practice interpreting parameters in trigonometric form, and use compass heading analogies. Watch for: confusion between magnitude and direction parameters, misidentifying which vector components are affected by which parameters.