Projection Math Example 2

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Example 2

hard
Vector a⃗=(3,4)\vec{a} = (3, 4) is projected onto vector b⃗=(1,0)\vec{b} = (1, 0). Find (a) the scalar projection and (b) the vector projection. Then find the projection onto c⃗=(1,1)/2\vec{c} = (1, 1)/\sqrt{2}.

Solution

  1. 1
    Step 1: Scalar projection of a⃗\vec{a} onto b^\hat{b} (unit vector): compb⃗a⃗=a⃗⋅b^=(3)(1)+(4)(0)=3\text{comp}_{\vec{b}}\vec{a} = \vec{a}\cdot\hat{b} = (3)(1)+(4)(0) = 3.
  2. 2
    Step 2: Vector projection: projb⃗a⃗=3b^=3(1,0)=(3,0)\text{proj}_{\vec{b}}\vec{a} = 3\hat{b} = 3(1,0) = (3, 0).
  3. 3
    Step 3: c^=(1,1)/2\hat{c} = (1,1)/\sqrt{2}. Scalar projection =a⃗⋅c^=3/2+4/2=7/2= \vec{a}\cdot\hat{c} = 3/\sqrt{2} + 4/\sqrt{2} = 7/\sqrt{2}.
  4. 4
    Step 4: Vector projection onto c⃗\vec{c}: 72⋅(1,1)2=72(1,1)=(3.5, 3.5)\dfrac{7}{\sqrt{2}} \cdot \dfrac{(1,1)}{\sqrt{2}} = \dfrac{7}{2}(1,1) = (3.5,\, 3.5).

Answer

Onto b⃗\vec{b}: scalar =3=3, vector =(3,0)=(3,0). Onto c⃗\vec{c}: scalar =7/2=7/\sqrt{2}, vector =(3.5,3.5)=(3.5, 3.5).
The scalar projection gives the signed length of the shadow; the vector projection is that length in the direction of the target. The formula proju^a⃗=(a⃗⋅u^)u^\text{proj}_{\hat{u}}\vec{a} = (\vec{a}\cdot\hat{u})\hat{u} works for any unit vector u^\hat{u}.

About Projection

The image formed when points of a shape are mapped onto a lower-dimensional surface along parallel or converging rays.

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