Scalar and Vector Projections

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Presentation transcript:

Scalar and Vector Projections Accelerated Math III February 1, 2016

One-Minute Question Evaluate: (3i - 7j ) •(4i + j )

Scalar Projections A scalar projection     of u onto v is the length of the shadow that   u casts on v. u v Scalar projection

Vector Projections A vector projection     of u onto v is the vector in the same direction as v, whose length is the scalar projection of   u on v. u v Vector Projection

Vector Projections A vector component of     u orthogonal to v is the vector perpendicular to v, that adds to the vector projection of   u on v to create u. u v Vector Projection Orthogonal Component

Now you try these!!

So What Happens If We Have Angles and Lengths? How long is the scalar projection of u onto v? Shouldn’t it be: |u|cos  if  is the angle between vectors

So try these! Find the scalar projection of r on s if |r| = 10, |s| = 16, and the angle between them is 140º. 10cos 140º = -7.66 Find the length of the orthogonal component of r on s. (-7.66)2 + |o|2 = 102 So |o| = 6.428

So What Happens If We Have Coordinates? How long is the scalar projection of u onto v? Shouldn’t it be: |u|cos  if  is the angle between vectors So isn’t that |u| (u • v) = (u • v) |u||v| |v|

Try This One! Find the scalar projection, the vector projection, and the orthogonal component of: (6i + 7j ) onto (5i - 12j ) The scalar projection is: 6·5 + 7·(-12) = -54/13 The vector projection is: -54(5i - 12j )/132 = -270i/169 + 648j/169 The orthogonal component is: (6i + 7j ) - (-270i/169 + 648j/169) = 1284i/169- 535j/169