Vector Intuition Math Example 4

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Example 4

hard
Find the unit vector in the direction of w⃗=(3,−4)\vec{w} = (3, -4).

Solution

  1. 1
    Step 1: Compute the magnitude: ∣w⃗∣=32+(−4)2=9+16=5|\vec{w}| = \sqrt{3^2 + (-4)^2} = \sqrt{9+16} = 5.
  2. 2
    Step 2: Divide each component by the magnitude: w^=15(3,−4)=(0.6,−0.8)\hat{w} = \dfrac{1}{5}(3, -4) = (0.6, -0.8).

Answer

w^=(0.6,−0.8)\hat{w} = (0.6, -0.8)
A unit vector has magnitude 1 and points in the same direction as the original vector. Dividing by the magnitude (normalising) preserves direction while setting the magnitude to 1. Unit vectors are used to represent pure direction, independent of magnitude.

About Vector Intuition

A mathematical object with both a magnitude (size) and a direction, often drawn as an arrow.

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More Vector Intuition Examples

Example 1 easy

A displacement vector points 3 units east and 4 units north. What is the magnitude of this vector?

Example 2 medium

Vectors [formula] and [formula]. Find [formula] and interpret the result geometrically.

Example 3 easy

What is the difference between a vector and a scalar? Give one example of each.

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