#AP Pre-Calculus: Vectors - Your Night-Before Review 🚀

Hey! Let's get you super confident about vectors for your AP Pre-Calculus exam. This guide is designed to be quick, clear, and exactly what you need right now. Let's dive in!

#1. Introduction to Vectors

#What are Vectors? 🏹

Vectors are like arrows that have both magnitude (length) and direction. Think of them as representing movement, force, or any quantity with a direction. They're not just numbers; they're like little guided missiles! 🎯

• Tail: Where the vector starts.
• Head: Where the vector ends.

#Visualizing Vectors

Imagine a vector as an arrow on a graph. The arrow's length is its magnitude, and the direction it points is its direction. Simple, right? ↗️

#2. Representing Vectors

#2D Vector Components 🛩️

In a 2D plane, we use components to define a vector. Think of it like giving directions: "move this much in the x-direction, and this much in the y-direction."

• Two Points: Defined by $P1 = (x1, y1)$ (tail) and $P2 = (x2, y2)$ (head).
• Components: $<a, b>$, where $a = x2 - x1$ and $b = y2 - y1$.

#Special Case: The Zero Vector

• $<0, 0>$: When the tail and head are the same point. It has zero magnitude. 🤓

#Trigonometric Components

We can also find vector components using trigonometry. If you know the angle a vector makes with the x-axis and its magnitude, you can find the x and y components using cosine and sine, respectively. 📐

#3. Vector Operations

#Scalar Multiplication ✖️

Multiplying a vector by a constant (scalar) changes its magnitude but not its direction (unless the scalar is negative, then it reverses the direction). Think of it as zooming in or out on the vector. 🦤

#Vector Addition ➕

Adding vectors is like combining movements. Place the tail of the second vector at the head of the first, and the resultant vector goes from the tail of the first to the head of the second. ✌️

• Component-wise Addition: Add corresponding components: $<a1, b1> + <a2, b2> = <a1 + a2, b1 + b2>$.
• Parallelogram Rule: Geometrically, it's the diagonal of the parallelogram formed by the two vectors. 🏖️

#Dot Product 🔘

The dot product is a way to multiply two vectors and get a scalar. It tells us how similar the vectors are in direction. 🔢

• Formula: $A · B = (A_x * B_x) + (A_y * B_y)$.
• Similarity:
  • 0: Vectors are perpendicular.
  • Positive: Vectors point in similar directions.
  • Negative: Vectors point in opposite directions.

#Unit Vectors ✔️

A unit vector has a magnitude of 1. It's like a direction pointer without any size. 1️⃣

• Finding a Unit Vector: Divide a vector by its magnitude. This "shrinks" the vector to a length of 1. 🤓

• Standard Unit Vectors:
  • i = $<1, 0>$ (x-axis)
  • j = $<0, 1>$ (y-axis)
• Linear Combination: Any vector $<a, b>$ can be written as $ai + bj$. 🚲

#4. Appendix: Side Lengths and Angles

#Law of Sines 📐

Relates the sides of a triangle to the sines of their opposite angles. Useful when you have two angles and a side, or two sides and an angle opposite one of them. 🫰

• Formula: $a/sin(A) = b/sin(B) = c/sin(C)$

#Law of Cosines 📐

Relates the sides of a triangle to the cosine of one of its angles. Useful when you have three sides, or two sides and the included angle. 🟰

• Formula:
  • $a² = b² + c² - 2bc * cos(A)$
  • $b² = a² + c² - 2ac * cos(B)$
  • $c² = a² + b² - 2ab * cos(C)$

#Triangle Inequality Theorem

• The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This helps determine if a set of measurements corresponds to a valid triangle. 📐

#5. Final Exam Focus

#High-Priority Topics

• Vector Components: Converting between geometric and component forms.
• Vector Operations: Addition, scalar multiplication, and dot product.
• Unit Vectors: Finding and using them.
• Law of Sines and Cosines: Applying them to vector problems.

#Common Question Types

• Finding the resultant vector after addition.
• Calculating the dot product and interpreting its meaning.
• Determining unit vectors.
• Solving triangles formed by vectors using the Law of Sines and Cosines.

#Last-Minute Tips

• Time Management: Don't spend too long on one question. Move on and come back if time allows.
• Common Pitfalls:
  • Confusing dot product with vector addition.
  • Forgetting to square the components when finding the magnitude.
  • Misapplying the Law of Sines and Cosines.
• Strategies:
  • Draw diagrams to visualize vectors.
  • Double-check your calculations.
  • If stuck, try a different approach.

#6. Practice Questions

Practice Question

#Multiple Choice Questions

1. Given vectors $\vec{u} = <2, -3>$ and $\vec{v} = <-1, 4>$, find $2\vec{u} - \vec{v}$. (A) $<5, -10>$ (B) $<3, 1>$ (C) $<5, -2>$ (D) $<3, -10>$

2. What is the dot product of $\vec{a} = <5, 2>$ and $\vec{b} = <-3, 4>$? (A) 2 (B) -7 (C) 23 (D) -15

3. Find a unit vector in the direction of $\vec{w} = <6, 8>$. (A) $<\frac{3}{5}, \frac{4}{5}>$ (B) $<\frac{6}{10}, \frac{8}{10}>$ (C) $<\frac{3}{10}, \frac{4}{10}>$ (D) $<\frac{5}{3}, \frac{5}{4}>$

#Free Response Question

Vectors $\vec{p}$ and $\vec{q}$ have magnitudes of 5 and 8, respectively, and the angle between them is 60 degrees.

(a) Find the dot product of $\vec{p}$ and $\vec{q}$. (2 points)

(b) Find the magnitude of the vector $\vec{p} + \vec{q}$. (4 points)

(c) Find the angle between $\vec{p}$ and $\vec{p} + \vec{q}$. (3 points)

Scoring Breakdown:

(a)

• 1 point: Correctly uses the formula for the dot product involving magnitudes and the angle between the vectors.
• 1 point: Correct calculation and answer. $||p|| ||q|| cos(60) = 5 * 8 * 0.5 = 20$

(b)

• 1 point: Recognizes the need to use the Law of Cosines or the component method.

• 2 points: Correctly sets up the Law of Cosines or adds vectors using components.

• 1 point: Correct answer. The magnitude of $\vec{p} + \vec{q}$ is approximately 11.4. (c)

• 1 point: Recognizes the need to use the dot product or the Law of Sines/Cosines.

• 1 point: Correctly sets up the equation to solve for the angle.

• 1 point: Correct answer. The angle between $\vec{p}$ and $\vec{p} + \vec{q}$ is approximately 26.3 degrees.

You've got this! Go ace that exam! 💪