Scalar and Vector Projections
Scalar Projections
When asked to find the scalar projection of one vector
onto another vector, you are being asked to calculate the
component of the first vector in the direction of the
second. The result is a SCALAR QUANTITY.
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“ the component of b; the scalar distance of ON;
scalar projection of a on b.
Notation:
Scalar projection of a on b:
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Vector Projection
The vector projection of a on b is the vector ON Since
ON will always be in the direction of b, we can
calculate the required vector projection by multiplying b
by a ratio of magnitude of ON and b.
unit vector in the direction of b
Notation: Vector projection of a on b:
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Example: Find the scalar and vector projections
for a = (2, 4, -1) on b = (3, 3, 4).
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Example: Calculate the scalar and vector projections
of a = (2, 3, - 4) on each coordinate axes.
Note: If OP is the position vector of the point P(a, b, c) ,
then the scalar projection of OP = (a, b, c) on the standard
basis vectors i, j, k and are a, b, and c respectively. The
vector projections of OP on the axes are ai, bj and ck.
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Direction Angles and Cosines
To find the angles a vector makes with
the coordinate axes:
(a, b, c)
Consider the vector OP = v = (a, b, c), the
direction angles of OP for
The positive x, y, z axes are commonly
denoted by α, β, γ respectively.
pg 398 #1,3,5­7,11, 14a, 15a
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