Direction Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Two vectors: aβƒ—=(1,0)\vec{a} = (1, 0) (east) and bβƒ—=(0,βˆ’1)\vec{b} = (0, -1) (south). What is the angle between them?

Solution

  1. 1
    Step 1: Use the dot product formula: cos⁑θ=aβƒ—β‹…bβƒ—βˆ£aβƒ—βˆ£βˆ£bβƒ—βˆ£\cos\theta = \dfrac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}.
  2. 2
    Step 2: aβƒ—β‹…bβƒ—=(1)(0)+(0)(βˆ’1)=0\vec{a} \cdot \vec{b} = (1)(0) + (0)(-1) = 0.
  3. 3
    Step 3: ∣aβƒ—βˆ£=1|\vec{a}| = 1, ∣bβƒ—βˆ£=1|\vec{b}| = 1.
  4. 4
    Step 4: cos⁑θ=0β‡’ΞΈ=90Β°\cos\theta = 0 \Rightarrow \theta = 90Β°.

Answer

The angle between them is 90Β°90Β°.
East and south are perpendicular directions β€” a 90Β° angle separates them. The dot product being zero is the algebraic signature of perpendicularity. This is consistent with intuition: a 90Β° clockwise rotation of east gives south.

About Direction

The orientation of movement or facing in space, independent of speed or distanceβ€”where something points.

Learn more about Direction β†’

More Direction Examples

Example 1 easy

A vector points from the origin to the point [formula]. What angle does it make with the positive x-

Example 3 easy

Do the vectors [formula] and [formula] point in the same direction? Explain.

Example 4 hard

A ship sails [formula] (30Β° east of north). Express this direction as a unit vector in the standard

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