1.3.E: The Cross Product (Exercises)

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Exercise \(\PageIndex{1}\)

For each of the following pairs of vectors \(\mathbf{x}\) and \(\mathbf{y}\), find \(\mathbf{x} \times \mathbf{y}\) and verify that \(\mathbf{x} \perp(\mathbf{x} \times \mathbf{y})\) and \(\mathbf{y} \perp(\mathbf{x} \times \mathbf{y})\).

(a) \(\mathbf{x}=(1,2,-1), \mathbf{y}=(-2,3,-1)\)

(b) \(\mathbf{x}=(-2,1,4), \mathbf{y}=(3,1,2)\)

(d) \(\mathbf{x}=(-1,4,1), \mathbf{y}=(3,2,-1)\)

Answer

(a) \(\mathbf{x} \times \mathbf{y}=(1,3,7)\)

(b) \(\mathbf{x} \times \mathbf{y}=(-2,16,-5)\)

(d) \(\mathbf{x} \times \mathbf{y}=(-6,2,-14)\)

Exercise \(\PageIndex{2}\)

Find the area of the parallelogram in \(\mathbb{R}^3\)that has the vectors \(\mathbf{x}=(2,3,1)\) and \(\mathbf{y}=(-3,3,1)\) for adjacent sides.

Exercise \(\PageIndex{3}\)

Find the area of the parallelogram in \(\mathbb{R}^2\)that has the vectors \(\mathbf{x}=(3,1)\) and \(\mathbf{y}=(1,4)\) for adjacent sides.

Answer: 11

Exercise \(\PageIndex{4}\)

Find the area of the parallelogram in \(\mathbb{R}^3\) that has vertices at (1, 1, 1), (2, 3, 2), (−2, 4, 4), and (−3, 2, 3).

Exercise \(\PageIndex{5}\)

Find the area of the parallelogram in \(\mathbb{R}^2\) that has vertices at (2,−1), (4,−2), (3,0), and (1,1).

Answer: 3

Exercise \(\PageIndex{6}\)

Find the area of the triangle in \(\mathbb{R}^3\) that has vertices at (1, 1, 0), (2, 3, 1), and (−1, 3, 2).

Exercise \(\PageIndex{7}\)

Find the area of the triangle in \(\mathbb{R}^2\) that has vertices at (−1, 2), (2, −1), and (1, 3).

Answer: \(\frac{9}{2}\)

Exercise \(\PageIndex{8}\)

Find the volume of the parallelepiped that has the vectors \(\mathbf{x}=(1,2,1)\), \(\mathbf{y}=(-1,1,1)\), and \(\mathbf{z}=(-1,-1,6)\) for adjacent sides.

Exercise \(\PageIndex{9}\)

A parallelepiped has base vertices at (1, 1, 1), (2, 3, 2), (−2, 4, 4), and (−3, 2, 3) and top vertices at (2, 2, 6), (3, 4, 7), (−1, 5, 9), and (−2, 3, 8). Find its volume.

Answer: 42

Exercise \(\PageIndex{10}\)

Verify the properties of the cross product stated in Equation (1.3.8) through (1.3.12).

Exercise \(\PageIndex{11}\)

Since \(\vert \mathbf{z} \cdot (\mathbf{x} \times \mathbf{y}) \vert\) \(\vert \mathbf{y} \cdot (\mathbf{z} \times \mathbf{x}) \vert \), and \(\vert \mathbf{x} \cdot (\mathbf{y} \times \mathbf{z}) \vert \) are all equal to the volume of a parallelepiped with adjacent edges \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\), they should all have the same value.

Show that in fact

\[ \mathbf{z} \cdot (\mathbf{x} \times \mathbf{y}) = \mathbf{y} \cdot (\mathbf{z} \times \mathbf{x}) = \mathbf{x} \cdot (\mathbf{y} \times \mathbf{z}) \nonumber \]

How do these compare with \( \mathbf{z} \cdot (\mathbf{y} \times \mathbf{z})\), \(\mathbf{y} \cdot (\mathbf{z} \times \mathbf{x})\), and \(\mathbf{x} \cdot (\mathbf{z} \times \mathbf{y} ) \)?

Exercise \(\PageIndex{12}\)

Suppose \(\mathbf{x}\) and \(\mathbf{y}\) are parallel vectors in\(\mathbb{R}^3\). Show directly from the definition of the cross product that \(\mathbf{x} \times \mathbf{y}=\mathbf{0}\).

Exercise \(\PageIndex{13}\)

Show by example that the cross product is not associative. That is, find vectors \(\mathbf{x}\), \(\mathbf{y}\), and \(\mathbf{z}\) such that \[\mathbf{x} \times (\mathbf{y} \times \mathbf{z}) \neq (\mathbf{x} \times \mathbf{y}) \times \mathbf{z} \nonumber . \]

Answer: For example, \(\mathbf{e}_{2} \times\left(\mathbf{e}_{2} \times \mathbf{e}_{3}\right)=-\mathbf{e}_{3}\), whereas \(\left(\mathbf{e}_{2} \times \mathbf{e}_{2}\right) \times \mathbf{e}_{3}=\mathbf{0}\).