Miscellaneous Exercise 7: Vectors in Plane

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Chapter Overview & SLOs

How do we differentiate directional physics metrics and establish algebraic vector balances? The miscellaneous exercise for Chapter 7 acts as a comprehensive capstone review, anchoring vector representations, component expansions, absolute magnitudes, and formal geometric proofs.

Scalar versus Vector Quantities:

• Scalar quantity: Completely specified by numeric magnitude and units alone (speed, distance, work)
• Vector quantity: Requires both quantitative magnitude and spatial direction (velocity, force, torque)

Position Vector Translations: For points $A$ (position vector $\vec{a}$) and $B$ (position vector $\vec{b}$):

• $\vec{AB} = \vec{b} - \vec{a}$ (Terminal minus Initial)
• Negative vector: if $\vec{OP} = [-6, 7]$, then $-\vec{OP} = [6, -7]$

Evaluating Absolute Magnitudes: For $\vec{u} = -5\hat{i} + 12\hat{j}$:

• $|\vec{u}| = \sqrt{(-5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ units

Matching Equal Vector Components: Given $\vec{a} = x\hat{i} + 2\hat{j}$ and $\vec{b} = 6\hat{i} - (x+y)\hat{j}$:

• Equate horizontal components: $x = 6$
• Equate vertical components: $2 = -(x+y)$
• Substitute $x=6$: $2 = -(6+y)$ → $2 = -6 - y$ → $y = -8$

Important Rule: Two vectors are equal if and only if their corresponding directional components match exactly.

These notes are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we differentiate between scalar and vector physical quantities based on the presence or absence of explicit spatial directions?
• How do we formulate displacement paths between two coordinate points by applying the terminal-minus-initial position vector law ($\vec{AB} = \vec{b}-\vec{a}$)?
• How do we compute the exact absolute scalar magnitude of two-dimensional vectors using the rectangular coordinate root-sum-of-squares formula?
• How do we solve for unknown algebraic variables by applying component-by-component equality properties to matching unit vectors?

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