Projection Math Example 4

Follow the full solution, then compare it with the other examples linked below.

hard

Project the vector

\vec{v} = (2, -1, 3)

onto the unit vector

\hat{u} = (0, 0, 1)

(the

z

-axis direction). Interpret the result geometrically.

1
Step 1: Scalar projection $= \vec{v} \cdot \hat{u} = (2)(0)+(-1)(0)+(3)(1) = 3$ .
2
Step 2: Vector projection $= 3\hat{u} = 3(0,0,1) = (0, 0, 3)$ .

Answer

Projection

= (0, 0, 3)

, the

z

-component of

\vec{v}

Projecting onto a coordinate axis extracts that coordinate. The projection of

(2,-1,3)

onto the

z

-axis gives

(0,0,3)

, showing that the

z

-component

=3

is the 'shadow' of the vector on the

z

-axis. The remaining part

(2,-1,0)

is perpendicular to the

z

-axis.

About Projection

The image formed when points of a shape are mapped onto a lower-dimensional surface along parallel or converging rays.

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Vector [formula] is projected onto vector [formula]. Find (a) the scalar projection and (b) the vect

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