Vectors

You’re lost in the woods and someone radios directions: “Walk 3 miles.” Useless — 3 miles in which direction? Now they say: “Walk 3 miles due north, then 4 miles due east.” That’s useful, because each instruction has both a size (distance) and a direction.

Those instructions are vectors. And by combining them, you can figure out the shortcut home: the straight-line distance is 5 miles (a 3-4-5 right triangle) heading northeast. Vectors are the language of physics — forces, velocities, displacements, electric fields, magnetic fields are all vectors. On the MCAT, you need to add them, break them into components, use dot and cross products, and (for magnetism) apply the right-hand rule. This section is the master reference for all of that.

Magnitude and Direction

Every vector has two properties:

• Magnitude - the “size” or “how much” (always positive)
• Direction - the “which way” (expressed as an angle or as compass directions)

A vector is often written in bold (A) or with an arrow above it. Its magnitude is written as |A| or simply A (not bold).

Component Form

Any 2D vector can be expressed as components along the x and y axes:

A = (Ax, Ay), where Ax = A cos θ and Ay = A sin θ.

The magnitude is recovered from components using the Pythagorean theorem:

Adding Vectors by Components

This is the most reliable method for vector addition and the one you should default to on the MCAT.

To add A + B:

1. Break each vector into x and y components
2. Add x-components: Rx = Ax + Bx
3. Add y-components: Ry = Ay + By
4. Find magnitude and direction of the resultant R = (Rx, Ry)

Example: A boat travels 30 m/s east and a current pushes 40 m/s north. What is the boat’s actual velocity?

• Rx = 30 + 0 = 30 m/s (east)
• Ry = 0 + 40 = 40 m/s (north)
• |R| = sqrt($30^{2}$ + $40^{2}$) = sqrt(900 + 1600) = sqrt(2500) = 50 m/s
• θ = arctan($\frac{40}{30}$) = arctan($\frac{4}{3}$) = 53 degrees north of east

Subtracting Vectors

To subtract B from A: reverse the direction of B (flip its sign), then add.

A - B = A + (-B) = (Ax - Bx, Ay - By)

This is important for finding changes in velocity (Δv = $v_{\text{final}}$ - $v_{\text{initial}}$) in momentum and circular motion problems.

The Dot Product (Scalar Product)

The dot product takes two vectors and returns a scalar (a number, not a vector).

Physical meaning: The dot product measures how much two vectors point in the same direction. It is maximized when vectors are parallel (cos 0 = 1) and zero when they are perpendicular (cos 90 = 0).

MCAT application: Work. W = F . d = Fd cos θ. The force component parallel to displacement does work. A force perpendicular to motion does zero work.

The Cross Product (Vector Product)

The cross product takes two vectors and returns a new vector perpendicular to both.

Physical meaning: The cross product measures how much two vectors are perpendicular to each other. It is maximized when vectors are perpendicular (sin 90 = 1) and zero when they are parallel (sin 0 = 0).

MCAT applications:

• Torque: τ = r x F (force perpendicular to the lever arm produces maximum torque)
• Magnetic force: F = qv x B (a charge moving perpendicular to a magnetic field feels maximum force)

The Right-Hand Rule

To find the direction of A x B:

1. Point your fingers in the direction of A
2. Curl them toward B (through the smaller angle)
3. Your thumb points in the direction of A x B

Important: A x B = -(B x A). The cross product is not commutative - order matters. Reversing the order flips the direction.