CLASSICAL-MECHANICS · Interactive Practice | Unit 2 · Video 1

Magnitude Meets Direction: The Rules Every Vector Obeys

IKey Formulas

Formula Name Description
C⃗=A⃗+B⃗\vec{C} = \vec{A} + \vec{B} Vector addition Head-to-tail or parallelogram rule
∣A⃗+B⃗∣≤∣A⃗∣+∣B⃗∣|\vec{A} + \vec{B}| \leq |\vec{A}| + |\vec{B}| Triangle inequality Magnitudes do not simply add
∣cA⃗∣=∣c∣ ∣A⃗∣|c\vec{A}| = |c|\,|\vec{A}| Scalar multiplication Stretches, shrinks, or flips a vector
A^=A⃗∣A⃗∣\hat{A} = \dfrac{\vec{A}}{|\vec{A}|} Unit vector Pure direction, length 1

IIInteractive Visualizations

Visualization 1 — Vector Addition

Joined head-to-tail, how does the length of C⃗=A⃗+B⃗\vec{C} = \vec{A} + \vec{B} compare with ∣A⃗∣+∣B⃗∣|\vec{A}| + |\vec{B}|?

Visualization 2 — Scalar Multiplication

Multiplying A⃗\vec{A} by a scalar cc stretches, shrinks, or flips it — but never rotates it off its own line.

Visualization 3 — The Unit Vector

Dividing A⃗\vec{A} by its own length gives A^=A⃗/∣A⃗∣\hat{A} = \vec{A}/|\vec{A}|, a vector of length exactly 1 in the same direction.

IIIQuiz Questions

Question 1

Which of the following physical quantities is a scalar (magnitude only, no direction)?

✅ Correct! Mass has magnitude only, with no direction — it is a scalar.

❌ Not quite. That quantity has a direction associated with it, which makes it a vector.

Show solution

Solution:

A scalar is fully described by a single number with units — it has magnitude only.

  • Mass (e.g., 5 kg) needs no direction, so it is a scalar.
  • Velocity, force, and displacement all have both magnitude and direction, so they are vectors.

The answer is Mass.

Question 2

Two forces act on a box. Force A⃗\vec{A} has magnitude 6 N and Force B⃗\vec{B} has magnitude 8 N, and they point in different (non-parallel) directions. What can we say about the magnitude of the resultant C⃗=A⃗+B⃗\vec{C} = \vec{A} + \vec{B}?

✅ Correct! Since the vectors aren't parallel, the magnitudes don't fully add.

❌ Not quite. Magnitudes only add to 14 N if the vectors point the same way. For different directions, the resultant is shorter.

Show solution

Solution:

By the triangle inequality: ∣A⃗+B⃗∣≤∣A⃗∣+∣B⃗∣=6+8=14 N|\vec{A} + \vec{B}| \leq |\vec{A}| + |\vec{B}| = 6 + 8 = 14 \text{ N}

Equality holds only when the two vectors point in exactly the same direction. Since they point in different directions here, the resultant is strictly shorter: ∣C⃗∣<14 N|\vec{C}| < 14 \text{ N}

Magnitudes do not simply add unless the vectors are parallel and same-pointing.

The answer is |C| < 14 N.

Question 3

A vector A⃗\vec{A} has magnitude 5. You compute the vector −2A⃗-2\vec{A}. Which statement correctly describes the result?

✅ Correct! |−2||A| = 10, and the negative sign reverses the direction.

❌ Not quite. Magnitude uses |c| so it is 10 (never negative), and a negative scalar flips direction without rotating.

Show solution

Solution:

The magnitude scales by the absolute value of the scalar: ∣−2A⃗∣=∣−2∣ ∣A⃗∣=2×5=10|-2\vec{A}| = |-2|\,|\vec{A}| = 2 \times 5 = 10

Magnitude is always non-negative, so it cannot be −10-10.

Since the scalar is negative, the direction is reversed (flipped to the opposite direction). A scalar never rotates a vector — it only stretches, shrinks, or flips it along the same line.

The answer is Magnitude 10, opposite direction to A.

Question 4

True or False: The unit vector of A⃗\vec{A}, defined as A^=A⃗/∣A⃗∣\hat{A} = \vec{A}/|\vec{A}|, always has magnitude 1 and points in the same direction as A⃗\vec{A}.

✅ Correct! Dividing by |A| (a positive scalar) gives length 1 and preserves direction.

❌ Not quite. |Â| = |A|/|A| = 1, and dividing by a positive scalar keeps the same direction. The statement is True.

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Solution:

Compute the magnitude of A^\hat{A}: ∣A^∣=∣A⃗∣A⃗∣∣=∣A⃗∣∣A⃗∣=1|\hat{A}| = \left|\frac{\vec{A}}{|\vec{A}|}\right| = \frac{|\vec{A}|}{|\vec{A}|} = 1

Dividing by ∣A⃗∣|\vec{A}| is multiplication by the positive scalar 1/∣A⃗∣1/|\vec{A}|, so the direction is unchanged. The unit vector strips away magnitude and keeps only direction — pure direction, length 1.

The statement is True.

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