Lesson 1: Vector Addition: The Head-to-Tail Method & Resultant Vectors

1. What is the difference between adding scalars and adding vectors?

A scalar only has magnitude (size), while a vector has magnitude AND direction.

Example: If you walk 3 meters East and 4 meters West, your total distance (scalar) is 3 + 4 = 7 meters. However, your displacement (vector) is 1 meter West. In physics, direction determines the sign and the math.

2. What does the "Head-to-Tail" method actually mean visually?

It is a way to draw vector addition. You draw the first vector (A). Then, you start the second vector (B) exactly where the first one ended (its "head"). You do not change the angle of the arrows.

The Resultant is the line drawn from the very beginning (the first tail) to the very end (the last head).

3. What is a "resultant" vector (R)?

The resultant is the "net" result. It is the single vector that produces the same effect as all the individual vectors combined. Think of it as the straight-line shortcut from your starting point to your final destination.

4. Is the magnitude |A + B| just the magnitude |A| plus the magnitude |B|?

No. This is only true if the vectors point in the exact same direction (angle θ = 0°).

If they are at angles, you must use geometry (like the Law of Cosines). The resultant is usually shorter than the sum of the individual lengths.

5. Why can't I just divide one vector by another vector (A / B)?

Division of vectors is undefined in standard physics. You can divide a vector by a scalar (which shrinks it), but dividing "North" by "East" has no mathematical meaning.

6. How do I subtract vectors for Relative Velocity problems (v_A - v_B)?

Treat subtraction as adding a negative.

A - B is the same as A + (-B).

To do this visually, take vector B, flip it 180° so it points in the opposite direction, and then add it to A using the Head-to-Tail method.

7. How do I resolve a vector into components if the angle isn't with the x-axis?

1. Don't blindly memorize "x is cosine, y is sine." Instead, look at the triangle:
2. Cosine is always the side adjacent (touching) the angle.
3. Sine is always the side opposite the angle.

Tip: On an inclined plane, gravity's components often flip (Force parallel to ramp = mg sin θ)

Lesson 2: Unit Vectors (i, j, k): How to Represent Direction in 3D Space

8. What exactly are î, ĵ, and k̂ hats?

They are specific labels for direction.

• î = 1 unit in the +x direction.
• ĵ = 1 unit in the +y direction.
• k̂ = 1 unit in the +z direction.

They act like "signposts" to tell you which way a number is pointing.

9. Why do we need unit vectors (û) if they have a value of 1?

Because multiplying by 1 doesn't change the size (magnitude) of a number, but sticking a unit vector next to it gives it a direction. It allows us to turn a scalar (like "50 mph") into a vector ("50 mph East") mathematically.

10. How do I find the unit vector of a specific vector?

You take the original vector and divide it by its total length (magnitude).

Formula: û = v / |v|

This "shrinks" the vector down so its length is exactly 1, but it keeps pointing the same way.

11. How do I calculate the magnitude of a vector in 3D space (x, y, z)?

Use the 3D Pythagorean Theorem. Square all the components, add them up, and take the square root.

Formula: |v| = √(x² + y² + z²)

Lesson 3: The Dot Product: Calculating Work and Scalar Values

12. If the Dot Product is zero (A · B = 0), does that mean one of the vectors is zero?

Not necessarily! It could mean the vectors are perpendicular (90° apart).

Since the Dot Product uses Cosine, and cos(90°) = 0, two strong forces pushing at right angles to each other have a zero Dot Product.

13. How do I find the angle θ between two vectors using the Dot Product?

You rearrange the standard formula.

Formula: θ = arccos( (A · B) / (|A| |B|) )

Step 1: Calculate the dot product. Step 2: Calculate the magnitudes. Step 3: Divide and take the inverse cosine.

14. When calculating Work, why do we use the Dot Product (F · d) instead of just multiplying force and distance?

Because only the part of the Force that points in the direction of motion actually does work. The Dot Product automatically filters out the perpendicular part of the force that is "wasted" (doesn't contribute to speed).

15. How do I find the projection of Vector A onto Vector B (proj_B A)?

You are essentially finding the "shadow" cast by A onto the line of B.

• Scalar Projection: Component = |A| cos θ
• Vector Projection: ( (A · B) / |B|² ) * B

16. Why is the Dot Product related to Cosine (cos θ)?

Cosine is maximum (1) when the angle is 0° (parallel). The Dot Product measures "parallel-ness." It is biggest when vectors point the same way and zero when they are perpendicular.

17. Why is the Dot Product a scalar but the Cross Product a vector?

• Dot Product: Asks "How much do these overlap?" The answer is an amount (Energy, Work), which is a scalar number.
• Cross Product: Asks "How much rotation do these create?" Rotation needs an axis to spin around, so the answer must be a vector (Torque)

Lesson 4: The Cross Product: Right-Hand Rule & Vector Direction

18. Does the order matter when I multiply vectors (A × B vs. B × A)?

YES. It is "Anti-Commutative."

• A × B = - (B × A).
• If you swap the order, the resulting vector points in the exact opposite direction (e.g., Up becomes Down).

19. How do I calculate the cross product using the matrix determinant method (ijk)?

Set up a 3x3 grid. Top row is î, ĵ, k̂. Second row is vector A. Third row is vector B.

Calculations:

• x-component: (AyBz - AzBy)
• y-component: -(AxBz - AzBx) <-- Don't forget the negative sign on the middle term!
• z-component: (AxBy - AyBx)

20. How do I use the Right-Hand Rule to find the direction of magnetic force or torque?

1. Point fingers straight in direction of the first vector (A).
2. Curl fingers toward the direction of the second vector (B).
3. Your thumb points in the direction of the Result (A × B).

Note: This only works with your Right Hand!

21. If I have a force in 3D (î, ĵ, k̂), how do I find the torque about the origin?

Torque is position (r) cross Force (F).

• Formula: τ = r × F.
• Use the matrix method. Ensure r (position) is in the second row and F (force) is in the third row.

22. Why is the Cross Product related to Sine (sin θ)?

Sine is maximum (1) when the angle is 90° (perpendicular). The Cross Product measures "perpendicular-ness" (how effective a force is at causing rotation). It is zero if the vectors are parallel.

23. Does the Cross Product exist in 2D?

Technically, the Cross Product is defined for 3D or 7D. However, in 2D physics, we pretend the vectors are on a flat sheet of paper and the result points straight up out of the paper (+z) or down into the paper (-z).

24. What is the physical meaning of the result of a Cross Product?

Geometrically, the magnitude |A × B| equals the Area of the parallelogram formed by the two vectors. Physically, it usually represents a rotational influence (like Torque or Angular Momentum).

25. Is the direction of the cross product 'real' or just a mathematical convention?

It is a convention. We could have all agreed to use a "Left-Hand Rule," and all the math would still work (just with opposite signs). Because the world standardized on the Right-Hand Rule, we call these "Pseudovectors" or "Axial Vectors.