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Chapter 5: Problem 19

For what values of \(c\) is \(\|c(1,2,3)\|=1 ?\)

Short Answer

Expert verified
The values of \(c\) that satisfy \(\|c(1,2,3)\|=1\) are \(c=\frac{1}{\sqrt{14}}\) and \(c=-\frac{1}{\sqrt{14}}\).

Step by step solution

01

Define the Vector

We have a scalar \(c\) multiplied by the vector (1, 2, 3) which gives us the vector \((c,c*2,c*3)\).
02

Apply the Norm Formula

Now we apply the formula for the norm or magnitude to this vector. This gives us: \(\sqrt{(c)^{2} + (2c)^{2} + (3c)^{2}}\).
03

Equate to 1 and Simplify

We need to find when this equals 1. So, we set our equation to 1 and simplify: \(1 = \sqrt{(c)^{2} + (2c)^{2} + (3c)^{2}}\). Squaring both sides to eliminate the square root gives \(1 = c^{2} + 4c^{2} + 9c^{2}\). Combining like terms gives us \(14c^{2} = 1\).
04

Solve for \(c\)

To isolate \(c\), we can divide by 14 on both sides, which gives \(c^{2}= \frac{1}{14}\). Finally, taking the square root of both sides gives that \(c=\frac{1}{\sqrt{14}}\) and \(c=-\frac{1}{\sqrt{14}}\).

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