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Vectors in 3-Dimensional Space Section 8-3

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WHAT YOU WILL LEARN: 1.How to add and subtract vectors in 3-dimensional space. 2.How to find the magnitude of vectors in 3- dimensional space.

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Vectors in Three-Dimensional Space Three Dimensional Vectors Locating Points in 3 Dimensions - Plot (-5, 3, 4) - Plot (3, -4, -2) x+ y+ z+ x- y- z-

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Ordered Triples Ordered triples can be used to represent vectors in three-dimensional space. They work just like two dimensions. Example: A directed line segment from the origin O to P (x, y, z) is called vector OP. The distance from the origin to point (x, y, z) is:

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The “ Mathy ” Stuff Suppose P 1 (x 1, y 1, z 1 ) is the initial point of a vector in space and P 2 (x 2, y 2, z 2 ) is the terminal point. The ordered triple that represents P 1 P 2 is: Its magnitude is given by: |P 1 P 2 | =

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Example Write the ordered triple that represents the vector from X(5, -3, 2) to Y(4, -5, 6). What is its magnitude?

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You Try Write the ordered triple that represents the vector from A(-2, -5, 0) to B(3, 1, 8). What is its magnitude?

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Operations on Vectors The operations we learned for two-dimensional vectors still hold true for three-dimensional vectors. Example: Find an ordered triple that represents 3p – 2p if p = and q =

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Unit Vectors in Three Dimensions We will add a new unit vector representation. We already have i for the x-axis, j for the y-axis and we add k for the z-axis. Example: Write AB as the sum of unit vectors for A(5, 10, -3) and B(-1, 4, -2).

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You Try Write GH as the sum of unit vectors for G(-2, -5, 4) and H(1, 5, 6)

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Homework page 503, even, 30, 32, 38

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