Q. 61
Question
Prove that vectors \(u\) and \(v\) are orthogonal if and only if \(u · v = 0\). (This is Theorem 10.19.)
Step-by-Step Solution
Verified Answer
It is proven that vectors \(u\) and \(v\) are orthogonal if and only if \(u · v = 0\).
1Step 1. Given Information
We have to prove that vectors \(u\) and \(v\) are orthogonal if and only if \(u · v = 0\).
Orthogonal vectors are those vectors that are perpendicular to each other.
2Step 2. Prove
To prove that vectors \(u\) and \(v\) are orthogonal if and only if \(u · v = 0\). We will use the formula for the angle between two vectors \(u\) and \(v\) which is \(u\cdot v=\left\|u \right\|\left\|v \right\|cos\theta\).
Since \(u\neq 0,v\neq 0\). So, \(u\cdot v=0 \) is possible only if \(cos\theta=0\).
Thus, \(u · v = 0\) if the vectors are orthogonal.
Other exercises in this chapter
Q. 57
Show that for any vector \(v\) in \(\mathbb{R}^{3}\),\(v = (v · i)i + (v · j)j + (v · k)k\).
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Let \(θ\) be the angle between nonzero vectors \(u\) and \(v\).Prove each of the following:(a) \(θ\) is acute if and only if \(u · v > 0\)
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Let \(u\) be a nonzero vector and let \(v\) be any vector. Show that the decomposition \(v=v_{\parallel }+v_{\perp }\), where \(v_{\parallel }\) is parallel to
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Use a vector argument to prove that a parallelogram is a rectangle if and only if the diagonals have the same length.
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