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Use the scalar triple product to verify that the vectors $ u = i + 5j - 2k $, $ v = 3i - j $, and $ w = 5i + 9j - 4k $ are coplanar.

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$$\begin{aligned} \mathbf{v} \cdot(\mathbf{u} \times \mathbf{w}) &=<3 .-1.0>\cdot(<1,5,-2>x<5,9,-4>) \\ &=<3,-1,0>\cdot(<-2,-6,-16>) \\ &=-6+6+0 \\ &=0 \end{aligned}$$

01:53

Wen Zheng

04:24

Willis James

Calculus 3

Chapter 12

Vectors and the Geometry of Space

Section 4

The Cross Product

Vectors

Johns Hopkins University

Campbell University

Baylor University

University of Michigan - Ann Arbor

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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Use the scalar triple prod…

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Three vectors $\mathbf{a},…

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Three vectors $\mathbf{u},…

Okay, there's a theorem and it says okay if the triple product okay equals zero the victors our complaint to. So all we have to do is get the triple product to equal to zero. And the triple product just means The Determinant 1 5 -2. 3 -10 5. -4. Okay so I'm not sure how you know how to do it, but here's one way you copy the first column, you copy the second column, Then you go diagonally. So 1 -1 -4. That's four plus five. plus -239 -6 times 9 -6 times nine. Yeah. Oh sorry minus 50 for minus now. Go the other way five minus 12. So positive 10 plus zero plus mm 15 times minus four minus 60. And so we get minus 50 for minus 50 minus minus 50 which equals zero. So completely.

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