Components and Projection u is a unit vector then the component of A in the direction of u u is If A is any vector and u u. A·u (Note: the component is a scalar.) u then since |u If θ is the angle between A and u u| = 1 u = |A||u A·u u| cos θ = |A| cos θ. The ﬁgure shows that geometrically this is the length of the leg of the right triangle with u. hypotenuse A and one leg parallel to u A θ u u |A| cos θ u the orthogonal projection of A on u u. We also call the leg parallel to u For a non-unit vector: the component of A in the direction of B is simply the component B u = |B| of A in the direction of u . (u u is the unit vector in the same direction as B.) Example: Find the component of A in the direction of B. i) |A| = 2, |B| = 5, θ = π/4. √ Answer: Referring to the ﬁgure above: the component is |A| cos θ = 2 cos(π/4) = 2. Note, the length of B given is irrelevant, since we only care about the unit vector parallel to B. ii) A = i + 2j, B = 3i + 4j. Answer: Unit vector in direction of B is 3/5 + 8/5 = 11/5. B |B| = 3 5i + 45 j ⇒ component is A · B/|B| = u = (−1, 0) iii) Find the component of A = (2, 2) in the direction of u u is a unit vector, so the component is A·u Answer: The vector u u = (2, 2)·(−1, 0) = −2. The u point in opposite directions. negative component is okay, it says the projection of A and u A θ u u We emphasize one more time that the component of a vector is a scalar.

Components and Projection

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