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Scalar and Vector Projections

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<strong>Scalar</strong> <strong>Projections</strong><br />

<strong>Scalar</strong> <strong>and</strong> <strong>Vector</strong> <strong>Projections</strong><br />

When asked to find the scalar projection of one vector<br />

onto another vector, you are being asked to calculate the<br />

component of the first vector in the direction of the<br />

second. The result is a SCALAR QUANTITY.<br />

1


“ the component of b; the scalar distance of ON;<br />

scalar projection of a on b.<br />

Notation: <strong>Scalar</strong> projection of a on b:<br />

2


<strong>Vector</strong> Projection<br />

The vector projection of a on b is the vector ON Since<br />

ON will always be in the direction of b, we can<br />

calculate the required vector projection by multiplying b<br />

by a ratio of magnitude of ON <strong>and</strong> b.<br />

unit vector in the direction of b<br />

Notation: <strong>Vector</strong> projection of a on b:<br />

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Example: Find the scalar <strong>and</strong> vector projections<br />

for a = (2, 4, -1) on b = (3, 3, 4).<br />

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Example: Calculate the scalar <strong>and</strong> vector projections<br />

of a = (2, 3, - 4) on each coordinate axes.<br />

Note: If OP is the position vector of the point P(a, b, c) ,<br />

then the scalar projection of OP = (a, b, c) on the st<strong>and</strong>ard<br />

basis vectors i, j, k <strong>and</strong> are a, b, <strong>and</strong> c respectively. The<br />

vector projections of OP on the axes are ai, bj <strong>and</strong> ck.<br />

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Direction Angles <strong>and</strong> Cosines<br />

To find the angles a vector makes with<br />

the coordinate axes:<br />

(a, b, c)<br />

Consider the vector OP = v = (a, b, c), the<br />

direction angles of OP for<br />

The positive x, y, z axes are commonly<br />

denoted by α, β, γ respectively.<br />

pg 398 #1,3,5­7,11, 14a, 15a<br />

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