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If $ r = \langle x, y, z \rangle $ , $ a = \langle a_1, a_2, a_3 \rangle $ and $ b = \langle b_1, b_2, b_3 \rangle $ show that the vector equation $ (r - a) \cdot (r - b) = 0 $ represents a sphere, and find its center and radius.

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Center $\left(\frac{a_{1}+b_{1}}{2}, \frac{a_{2}+b_{2}}{2}, \frac{a_{3}+b_{3}}{2}\right) ;$ radius $\frac{|\mathbf{a}-\mathbf{b}|}{2}$

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 3

The Dot Product

Vectors

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Lectures

02:56

In mathematics, a vector (from the Latin word "vehere" meaning "to carry") is a geometric entity that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. Vectors play an important role in physics, engineering, and mathematics.

11:08

In mathematics, a vector (from the Latin word "vehere" which means "to carry") is a geometric object that has a magnitude (or length) and direction. A vector can be thought of as an arrow in Euclidean space, drawn from the origin of the space to a point, and denoted by a letter. The magnitude of the vector is the distance from the origin to the point, and the direction is the angle between the direction of the vector and the axis, measured counterclockwise.

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01:58

The problem is: if r is equal to x y c, a is equal to a 1 a 2 a 3 and b is equal to b 1 b 2 b 3 showed that the vector equation, r minus a dot r minus b is equal to 0, represent A figure and find its sin to and writers the first r minus a is equal to x, minus 81 and i minus 82, and that t minus a 3 r minus b is equal to x, minus p, 1 y minus b, 2 t minus b 3 and Fitting r minus a dot r minus b is equal to 0 point, so we have x minus a 1 times x, minus 1 plus y minus 82 times y minus b, 2 plus c minus 83 times t minus b, 3 is equal to 0 and then we Have this is x, squared minus a 1 plus b 1 x, plus a 1 b 1 plus y square minus 82 plus 2 times y plus 82 b, 2 plus c square minus a 3 plus 3 times t plus 83 p 3 is equal to 0 point. Then we complete the squares. We have x minus a 1 plus this 1 plus b 1 square plus y minus 82 plus v 2 over 2 square plus c minus a 3 plus 3 over 2 square is equal to a 1 plus b 1 squared over 4 plus a 2 plus b 2 square over 4 plus 83 plus 83 squared over 4 minus a 1 b, 1 minus 82 v, 2 minus 83 v 3. This is equal to la force, times 1 minus v, 1 square plus 82 minus b, 2 square plus 83 minus b, 3 square. So to work, the equation: r, minus 8 dot r minus b is equal to 0 represents a sphere, and the center is a 1 plus b 1 over 2, a 2 plus b 2 over 2, a 3 plus e 3 over 2 and various is equal to 1: half root of a 1 minus b 1 squared plus a 2 minus b, 2 square plus 83 minus b 3 square.

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