AP Precalculus: Vectors

A vector-valued function gives a vector for every input. The most important use is planar motion: a particle with parametric equations x(t), y(t) has a position vector p(t) = ⟨x(t), y(t)⟩ = x(t)i + y(t)j. The signs tell you the location: x(t) > 0 means to the right of the origin, y(t) > 0 means above it. The magnitude of the position vector, |p(t)| = √(x(t)² + y(t)²), is the distance of the particle from the origin at time t. A velocity vector v(t) = ⟨v_x(t), v_y(t)⟩ describes how the particle is moving at time t: the sign of v_x says whether it is moving right (+) or left (−), and the sign of v_y says whether it is moving up (+) or down (−). The speed is the magnitude of the velocity, |v(t)| = √(v_x² + v_y²), which is a scalar and is never negative. Do not mix up the two magnitudes: |p(t)| measures how far the particle is from the origin, while |v(t)| measures how fast it is moving. Include units, such as meters and seconds, whenever the problem gives them.

1. Vectors: Scalars, Component Form, and Magnitude

A scalar is a quantity described by a single number: 40 miles per hour, 12 liters, 25°C. A vector needs two pieces of information, a magnitude (how much) and a direction (which way). “40 miles per hour” is a scalar speed, but “40 miles per hour due north” is a vector velocity. Force, displacement, and wind are all vectors. A vector is drawn as an arrow: the starting point is the tail, the end with the arrowhead is the head, and the length of the arrow is the magnitude. Only the length and direction matter, so a vector can be slid anywhere without changing it. To compute with vectors, write them in component form v = ⟨a, b⟩, where a is the horizontal change and b is the vertical change from tail to head. If the tail is (x₁, y₁) and the head is (x₂, y₂), then v = ⟨x₂ − x₁, y₂ − y₁⟩: always head minus tail. The magnitude (also called length or norm) is written |v| and comes from the Pythagorean theorem: |v| = √(a² + b²).

2. Equal, Opposite, and Parallel Vectors

Two vectors are equal when they have the same magnitude and the same direction, no matter where they are drawn. In component form this is simple: ⟨a, b⟩ = ⟨c, d⟩ exactly when a = c and b = d. So two arrows in different places are equal as long as head minus tail gives the same components. The negative of a vector reverses its direction and keeps its length: −⟨a, b⟩ = ⟨−a, −b⟩. The zero vector ⟨0, 0⟩ has no length and no direction. Vectors that point in the same or exactly opposite directions are parallel; algebraically, u is parallel to v when u = k·v for some nonzero scalar k, so the components are proportional. Setting two component forms equal is a quick way to solve for unknown numbers, because it produces one equation for each component.

3. Adding Vectors: Geometric and Algebraic

There are two equivalent ways to add vectors. Geometrically (head to tail): draw the first vector, slide the second so its tail touches the head of the first, and then the sum is the vector from the tail of the first to the head of the second. This is the triangle method. If both vectors start at the same point, complete the parallelogram and the sum is its diagonal from that point, the parallelogram method. Algebraically: add the matching components, ⟨a, b⟩ + ⟨c, d⟩ = ⟨a + c, b + d⟩. Vector addition is commutative (a + b = b + a) and associative. Adding the zero vector changes nothing, and adding a vector to its negative gives the zero vector. A useful shortcut for points: AB + BC = AC, because going from A to B and then B to C is the same as going from A to C. Be careful with magnitudes: |a + b| is usually not |a| + |b|, which is only true when the vectors point the same way.

4. Force and Wind Problems: Resultants with the Law of Cosines and Sines

Many real problems give vectors by magnitude and the angle between them instead of components. Two forces pulling on the same object, or a plane’s own velocity combined with a wind, can be replaced by one resultant vector, their sum. Sketch the two vectors from the same point and complete the parallelogram. Cut it along the diagonal and you get a triangle whose two known sides are the original vectors. If the angle between the vectors when their tails touch is θ, then the angle of the triangle opposite the resultant is the supplement 180° − θ, because the head-to-tail copy of the second vector turns the interior angle into its supplement. Magnitude comes from the law of cosines: R² = a² + b² − 2ab·cos(180° − θ), which is the same as R² = a² + b² + 2ab·cos θ. Direction comes from the law of sines: sin φ / (opposite side) = sin(180° − θ) / R, where φ is the angle between the resultant and one of the forces. When the vectors are perpendicular, skip the laws and use the Pythagorean theorem. Always check that the answer is reasonable: the resultant must be shorter than the sum of the two lengths and longer than their difference.

5. Scalar Multiplication

Multiplying a vector by a scalar k scales its length and, if k is negative, flips its direction: k⟨a, b⟩ = ⟨ka, kb⟩. The length changes by the factor |k|, so |kv| = |k|·|v|. If k > 1 the vector stretches; if 0 < k < 1 it shrinks; if k < 0 it also points the opposite way; and if k = 0 it becomes the zero vector. A vector and any scalar multiple of it are parallel. This gives a powerful way to build a vector with a prescribed direction and length: first make a vector of length 1 in the wanted direction by dividing by the magnitude, then multiply by the desired length. To get a vector opposite to v with length L, use −L·v/|v|. Scalar multiplication distributes over addition: k(a + b) = ka + kb, and (k + m)a = ka + ma.

6. Subtracting Vectors

Subtracting a vector means adding its negative: a − b = a + (−b). In components, subtract the matching entries: ⟨a, b⟩ − ⟨c, d⟩ = ⟨a − c, b − d⟩. Geometrically there are two pictures. Triangle form: draw a, then draw −b starting at the head of a; the arrow from the tail of a to the head of −b is a − b. Tails together: if a and b start at the same point, a − b is the arrow that goes from the head of b to the head of a. Memorize that phrase, “to the first, from the second,”, because it is the quickest way to draw a difference and it is exactly why a vector between two points is head minus tail. Subtraction is not commutative: b − a = −(a − b), which has the same length and the opposite direction. Combining operations follows ordinary algebra: 2a − 3b means scale first, then subtract.

7. The Scalar (Dot) Product

The dot product (or scalar product) of a = ⟨a₁, a₂⟩ and b = ⟨b₁, b₂⟩ is a · b = a₁b₁ + a₂b₂. Multiply matching components and add. The result is a single number, not a vector, which is why it is called the scalar product, and why it is different from scalar multiplication (a number times a vector gives a vector). Four properties are worth knowing: (1) a · b = b · a; (2) a · a = |a|²; (3) a · (b + c) = a · b + a · c; (4) (ka) · b = k(a · b). The second one is a handy way to get a magnitude: |a| = √(a · a). Because the dot product is a sum of products, it can be expanded like ordinary multiplication, so (a + b) · (c + d) = a · c + a · d + b · c + b · d. Do the dot product of vectors only; a dot product with a number such as 3 · a is not defined (that is scalar multiplication).

8. Geometric Properties of the Dot Product and the Angle Between Vectors

The dot product also has a geometric meaning: a · b = |a||b| cos θ, where θ (0° ≤ θ ≤ 180°) is the angle between the vectors when they are drawn from the same point. It comes from the law of cosines applied to the triangle formed by a, b, and b − a: |b − a|² = |a|² + |b|² − 2|a||b| cos θ. Because |a||b| is positive, the sign of a · b tells you the type of angle: a · b > 0 means θ is acute, a · b < 0 means θ is obtuse, and a · b = 0 means the nonzero vectors are perpendicular (θ = 90°). For parallel vectors, a · b = ±|a||b|, with + when they point the same way and − when opposite. To find the angle, solve for it: cos θ = (a · b) / (|a||b|) and θ = arccos of that number. Compute the dot product and both magnitudes separately, then combine. Keep your calculator in degree mode if you want degrees, and check the sign: a negative dot product must give an angle above 90°.

9. Unit Vectors and the i, j Form

A unit vector is any vector of length exactly 1. To check, compute the magnitude; to make one, divide a nonzero vector by its own length: u = v / |v|. This unit vector points in the same direction as v. If a component is unknown, set the magnitude equal to 1, square both sides, and solve; expect two solutions (a ± sign) unless the problem says which side. The two special unit vectors are i = ⟨1, 0⟩ and j = ⟨0, 1⟩, pointing along the positive x-axis and y-axis. Every vector can be built from them: ⟨a, b⟩ = ai + bj. The expression ai + bj is called unit vector form, and the magnitude formula is the same, |ai + bj| = √(a² + b²). Moving between the two notations is just reading off the coefficients. Unit vectors are the key to scaling: once you have the unit vector for a direction, multiplying by any positive length L gives a vector of length L in that direction.

10. Finding a Vector from Its Direction and Magnitude

Real situations rarely give components. A pilot reports a speed and a heading; a wind has a speed and a direction. To turn magnitude and direction into components, note that the unit vector at angle θ (measured from the positive x-axis) is ⟨cos θ, sin θ⟩, which is a point on the unit circle. Scaling by the magnitude gives v = |v|⟨cos θ, sin θ⟩ = |v|cos θ i + |v|sin θ j. In the right triangle formed by the vector and its horizontal and vertical legs, the horizontal component is the adjacent side and the vertical component is the opposite side. Going the other way, from components to direction, use tan θ = b/a, but choose the quadrant carefully: the arctangent gives an angle in quadrant I or IV only, so add 180° when a < 0. Always sketch the vector first. Directions measured as compass bearings (clockwise from north) must be converted to this standard angle before using the formula.

11. Vector-Valued Functions: Position, Velocity, and Speed

A vector-valued function gives a vector for every input. The most important use is planar motion: a particle with parametric equations x(t), y(t) has a position vector p(t) = ⟨x(t), y(t)⟩ = x(t)i + y(t)j. The signs tell you the location: x(t) > 0 means to the right of the origin, y(t) > 0 means above it. The magnitude of the position vector, |p(t)| = √(x(t)² + y(t)²), is the distance of the particle from the origin at time t. A velocity vector v(t) = ⟨v_x(t), v_y(t)⟩ describes how the particle is moving at time t: the sign of v_x says whether it is moving right (+) or left (−), and the sign of v_y says whether it is moving up (+) or down (−). The speed is the magnitude of the velocity, |v(t)| = √(v_x² + v_y²), which is a scalar and is never negative. Do not mix up the two magnitudes: |p(t)| measures how far the particle is from the origin, while |v(t)| measures how fast it is moving. Include units, such as meters and seconds, whenever the problem gives them.

Common Mistakes to Avoid

• Computing a vector between points as tail minus head. The component form is always head minus tail; reversing it gives the opposite vector.
• Adding magnitudes when adding vectors. |a + b| equals |a| + |b| only when the vectors point in the same direction; otherwise add the components first, then find the magnitude.
• Using the interior angle of the triangle with the wrong sign in the law of cosines. With the angle between the vectors when tails touch, use R² = a² + b² + 2ab cos θ (or the minus form with 180° − θ).
• Forgetting that a negative scalar reverses direction. Also, |kv| = |k||v|, so the magnitude never becomes negative.
• Confusing the dot product with scalar multiplication. a · b is a number; k·a is a vector.
• Reading the sign of the dot product backwards. Positive means an acute angle, negative means obtuse, and zero means perpendicular.
• Giving a unit vector that is not divided by the magnitude, or forgetting that making a vector of length L means L times the unit vector.
• Mixing up distance from the origin (|p(t)|, the magnitude of position) with speed (|v(t)|, the magnitude of velocity).

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