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Vector-Valued Functions

A vector has a length and a direction and is written in components, like ⟨3, 2⟩. A vector-valued function p(t) = ⟨x(t), y(t)⟩ gives a position vector for every value of t, so its tip traces a path. The velocity vector ⟨x′(t), y′(t)⟩ points along the path, and its length is the speed.

🎬 Step-by-step story

  1. A vector is an arrow with a length and a direction. ⟨3, 2⟩ means 3 right and 2 up; its length is √13 ≈ 3.61.
  2. To add vectors, put them tip to tail and add the parts. To scale a vector, multiply each part by the number.
  3. A vector-valued function p(t) = ⟨x(t), y(t)⟩ gives one position arrow for each t. The tip draws a path.
  4. x(t) and y(t) are two separate functions. Watch their shadows move on the two axes.
  5. The velocity vector ⟨x′(t), y′(t)⟩ touches the path. Its length is the speed.
  6. Free play: pick a path and slide t. Read p(t), velocity and speed below.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Is ⟨3, 2⟩ a point or a vector?

⟨3, 2⟩ is a move: 3 right, 2 up. It can start anywhere. When its tail sits at the origin, its tip is at the point (3, 2). That is why a position vector names a point.

Why is |u + v| not equal to |u| + |v|?

Lengths add only when the arrows point the same way. Tip-to-tail arrows that turn make a shortcut, so the sum is shorter.

How is a vector-valued function different from a parametric curve?

It is the same path. The vector view just draws an arrow from the origin to each point, so we can add, scale and measure it.

Why look at x(t) and y(t) separately?

Each part tells you one thing: x(t) says left or right, y(t) says up or down. The two shadows on the axes show this.

Can the speed be constant while the velocity changes?

Yes. On a circle the arrow length stays the same but the arrow keeps turning, so the velocity vector changes.

What does the velocity look like on a straight line path?

On ⟨−3 + 2t, −2 + t⟩ the velocity is ⟨2, 1⟩ at every t: same length and same direction.

Vectors in component form

A vector is a quantity with size and direction. We draw it as an arrow. In the plane we write it in component form: v = ⟨a, b⟩ means a steps along x and b steps along y. The same vector can be written ai + bj, where i = ⟨1, 0⟩ and j = ⟨0, 1⟩.

A vector from point A(x₁, y₁) to point B(x₂, y₂) is ⟨x₂ − x₁, y₂ − y₁⟩ (end minus start).

Adding, scaling and the dot product

Add: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩. In the picture, put the second arrow's tail on the first arrow's tip.

Scale: k⟨a, b⟩ = ⟨ka, kb⟩. If k is negative, the arrow flips round.

Dot product: ⟨a, b⟩ · ⟨c, d⟩ = ac + bd. It is a number, not a vector. It also equals |u||v| cos θ, where θ is the angle between the vectors. So:

Vector-valued functions and parametric paths

A vector-valued function takes an input t (often time) and gives a vector: p(t) = ⟨x(t), y(t)⟩. Place each output with its tail at the origin. Then its tip is the point (x(t), y(t)). As t changes, the tip traces a curve. This is the same curve as the parametric equations x = x(t), y = y(t).

You can study each part on its own: when is x(t) increasing? When is y(t) zero? Those answers tell you when the point moves right, or crosses the x-axis.

Examples: p(t) = ⟨3 cos t, 3 sin t⟩ is a circle of radius 3; p(t) = ⟨t, t²⟩ is a parabola; p(t) = ⟨1 + 2t, 3 − t⟩ is a straight line.

Velocity, speed and average rate of change

The average rate of change of p from t = a to t = b is the vector (p(b) − p(a)) ÷ (b − a): it points from the old position to the new one.

Shrink the gap and you get the velocity vector v(t) = ⟨x′(t), y′(t)⟩. It is tangent to the path and points the way the object is moving.

Speed = |v(t)| = √(x′(t)² + y′(t)²). Speed is a number; velocity is a vector.

Try it: In free play pick the circle. Move t. The speed stays 3, but the orange velocity arrow keeps turning. Now pick the parabola: the speed grows as the point climbs the side.

Key formulas and definitions

Worked examples

1. Find the magnitude and direction angle of v = ⟨3, 4⟩.

|v| = √(9 + 16) = √25 = 5. tan θ = 4/3, so θ ≈ 53.1° (first quadrant, both parts positive).

2. u = ⟨2, −1⟩, v = ⟨−3, 5⟩. Find 2u + v.

2u = ⟨4, −2⟩. 2u + v = ⟨4 − 3, −2 + 5⟩ = ⟨1, 3⟩.

3. Find the unit vector in the direction of ⟨−6, 8⟩.

Length = √(36 + 64) = 10. Unit vector = ⟨−6/10, 8/10⟩ = ⟨−0.6, 0.8⟩.

4. Find the angle between u = ⟨1, 2⟩ and v = ⟨4, −2⟩.

u · v = 4 − 4 = 0, so cos θ = 0 and θ = 90°. The vectors are perpendicular.

5. p(t) = ⟨t², 2t⟩. Find the position at t = 3 and the average rate of change from t = 1 to t = 3.

p(3) = ⟨9, 6⟩, p(1) = ⟨1, 2⟩. Average rate = (⟨9, 6⟩ − ⟨1, 2⟩) ÷ 2 = ⟨8, 4⟩ ÷ 2 = ⟨4, 2⟩.

6. p(t) = ⟨3 cos t, 3 sin t⟩. Find the velocity and speed at any t.

v(t) = ⟨−3 sin t, 3 cos t⟩. Speed = √(9 sin²t + 9 cos²t) = √9 = 3. The speed is constant even though the direction changes.

Common mistakes

Practice quiz

1. The magnitude of ⟨5, 12⟩ is:
2. ⟨2, 3⟩ + ⟨−1, 4⟩ equals:
3. If u · v = 0 (and neither is zero), the vectors are:
4. For p(t) = ⟨x(t), y(t)⟩, the velocity vector is:
5. p(t) = ⟨4 cos t, 4 sin t⟩ traces:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

What is a vector-valued function in simple words?

A rule that gives an arrow (a vector) for every input t. If the arrow starts at the origin, its tip moves along a path as t changes.

How do you find speed from a velocity vector?

Take the length of the velocity vector: speed = √(x′(t)² + y′(t)²).

What is the dot product used for?

To find the angle between two vectors, and to test if they are perpendicular (dot product 0).

Where this is taught

USA (Common Core, NGSS, AP)Grade 12Functions Involving Parameters, Vectors, and Matrices

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