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Suppose that $u_1$ and $u_2$ are orthogonal vectors, with $||u_1|| = 2$ and $||u_2|| = 5$. Find $||3u_1 + 4u_2||$ $$$$ Then, $u_1 \cdot u_1 = 4$ and $u_2\cdot u_2 = 25$. And $||3u_1 + 4u_2|| = \sqrt{(3u_1 + 4u_2)\cdot(3u_1 + 4u_2)}$ But the thing is I don't know what to do with this information. Any help?

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3 Answers 3

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Hint
For orthogonal vectors $u,v$ (this is defined as $u\cdot v = 0$) we have $$\| \alpha u + \beta v \|^2 = \alpha^2 \|u\|^2 + \beta^2 \|v\|^2$$

To see this, observe that $\alpha u$ and $\beta v$ will be orthogonal as well, so WLOG $\alpha=\beta=1$ and then $$\| u+v\|^2 = (u+v)\cdot(u+v) = u\cdot u + 2u\cdot v + v\cdot v = \|u\|^2 + 2\cdot 0 + \|v\|^2$$

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Hint: If $u_1, u_2$ are orthogonal, then their dot product is zero.

Can you take it from there?

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$$\|3u_1+4u_2\|^2=(3u_1+4u_2)\cdot(3u_1+4u_2)=9\|u_1\|^2+24u_1\cdot u_2+16\|u_2\|^2$$

What is $u_1\cdot u_2$?

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