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Exercise 3.4

Express the component of v ⟶ perpendicular to w ⟶ in terms of dot products.

Solution:

The component of v ⟶ perpendicular to w ⟶ is the magnitude of the projection of v ⟶ perpendicular to w ⟶ . By definition this projection is v ⟶ minus the projection of v ⟶ in the direction of w ⟶ .
But the projection of v ⟶ in the direction of w ⟶ is ( v ⟶ , w ⟶ ) ( w ⟶ , w ⟶ ) w ⟶ .
The projection of v ⟶ normal to w ⟶ is then v ⟶ − ( v ⟶ , w ⟶ ) ( w ⟶ , w ⟶ ) w ⟶ , the component of v ⟶ normal to w ⟶ is then the square root of the dot product of this projection with itself, or ( ( v ⟶ , v ⟶ ) − ( v ⟶ , w ⟶ ) * ( w ⟶ , v ⟶ ) ( w ⟶ , w ⟶ ) ) 1 / 2 .

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