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Chapter 1: Problem 9

Find \(k\) so that \(u\) and \(v\) are orthogonal, where: (a) \(u=(1, k,-3)\) and \(v=(2,-5,4)\) (b) \(u=(2,3 k,-4,1,5)\) and \(v=(6,-1,3,7,2 k)\)

Short Answer

Expert verified
For the given vectors, \(k = -2\) in case (a), and \(k = -1\) in case (b) to make the vectors orthogonal.

Step by step solution

01

(a) Finding k for u = (1, k, -3) and v = (2, -5, 4)

Recall that the dot product of two vectors is given by: \[u \cdot v = u_1v_1 + u_2v_2 + u_3v_3\] If \(u\) and \(v\) are orthogonal, then their dot product should be equal to 0: \[u \cdot v = 0\] Let's calculate the dot product of the given vectors \(u\) and \(v\) and set it equal to 0: \[(1)(2) + (k)(-5) + (-3)(4) = 0\] Now solve the equation for \(k\): \begin{align*} 2 - 5k - 12 &= 0 \\ -5k &= 10 \\ k &= -2 \end{align*} Therefore, the value of \(k\) that makes \(u\) and \(v\) orthogonal is -2.
02

(b) Finding k for u = (2,3k,-4,1,5) and v = (6,-1,3,7,2k)

We will again use the dot product to check for orthogonality. Let's calculate the dot product of the given vectors \(u\) and \(v\) and set it equal to 0: \[(2)(6) + (3k)(-1) + (-4)(3) + (1)(7) + (5)(2k) = 0\] Now solve the equation for \(k\): \begin{align*} 12 - 3k - 12 + 7 + 10k &= 0 \\ 7k &= -7 \\ k &= -1 \end{align*} Therefore, the value of \(k\) that makes \(u\) and \(v\) orthogonal is -1.

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