91Ó°ÊÓ

Verify the Pythagorean Theorem for the vectors \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u}=(1,-1), \quad \mathbf{v}=(1,1)\)

Verify the Pythagorean Theorem for the vectors \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u}=(3,-2), \quad \mathbf{v}=(4,6)\)

Verify the Pythagorean Theorem for the vectors \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u}=(3,4,-2), \quad \mathbf{v}=(4,-3,0)\)

Verify the Pythagorean Theorem for the vectors \(\mathbf{u}\) and \(\mathbf{v}\). \(\mathbf{u}=(4,1,-5), \quad \mathbf{v}=(2,-3,1)\)

Explain what is known about \(\theta\), the angle between \(\mathbf{u}\) and \(\mathbf{v},\) if (a) \(\mathbf{u} \cdot \mathbf{v}=0\) (b) \(\mathbf{u} \cdot \mathbf{v}>0\) (c) \(\mathbf{u} \cdot \mathbf{v}<0\)

Let \(\mathbf{v}=\left(v_{1}, v_{2}\right)\) be a vector in \(R^{2}\). Show that \(\left(v_{2},-v_{1}\right)\) is orthogonal to \(\mathbf{v},\) and use this fact to find two unit vectors orthogonal to the given vector. \(\mathbf{v}=(12,5)\)

Let \(\mathbf{v}=\left(v_{1}, v_{2}\right)\) be a vector in \(R^{2}\). Show that \(\left(v_{2},-v_{1}\right)\) is orthogonal to \(\mathbf{v},\) and use this fact to find two unit vectors orthogonal to the given vector. \(\mathbf{v}=(8,15)\)

Find the angle between the diagonal of a cube and one of its edges.

Find the angle between the diagonal of a cube and the diagonal of one of its sides.

Prove that if \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in \(R^{n},\) then \((\mathbf{u}+\mathbf{v}) \cdot \mathbf{w}=\) \(\mathbf{u} \cdot \mathbf{w}+\mathbf{v} \cdot \mathbf{w}\).