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11.2 Vectors in Space

11.2 Vectors in Space. Coordinates in Space. A three-dimensional coordinate system consists of: 3 axes: x-axis, y-axis and z-axis 3 coordinate planes: xy -plane, xz -plane and yz -plane 8 octants. Each point is represented by an ordered triple. z. Each vector is represented by. y.

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11.2 Vectors in Space

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  1. 11.2 Vectors in Space

  2. Coordinates in Space • A three-dimensional coordinate systemconsists of: • 3 axes: x-axis, y-axis and z-axis • 3 coordinate planes: xy-plane, xz-plane and yz-plane • 8 octants. Each point is represented by an ordered triple z Each vector is represented by y x

  3. Given 2 points in the space with coordinates Midpoint Formula: Distance Formula: Standard equation of a sphereof radius r, centered at is

  4. Examples 1) Find the standard equation of the sphere whose endpoints of a diameter are 2) Find the center and radius of the sphere: 3) Describe the solid satisfying the condition:

  5. Vectors The magnitudeof is: then v is a zero vector : If then v is a unit vector. If and are called the standard unit vectors.

  6. Vector Operations Vector sum: Scalar Multiplication: Negative (opposite): Vector difference • Vector v is parallel to u if and only if v = ku for some k.

  7. Examples (Normalize v) 1) Find the unit vector in the direction of v 2) Determine whether the points are collinear: 3) Show that the following points form the vertices of a parallelogram:

  8. Linear Combination Standard unit vector notation • v is called a linear combinationof i, j and k • Unit vector in the direction of v is given by • This unit vector is called the normalized form of v .

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