7.3: The Dot Product
- Page ID
- 143596
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- Let vector \(\mathbf{u} = (2, -3)\) and vector \(\mathbf{v} = (1, -4)\). Find \(\mathbf{u} \cdot \mathbf{v}\).
- Let vector \(\mathbf{a} = 3\mathbf{i} - 7\mathbf{j}\) and vector \(\mathbf{b} = -2\mathbf{j}\). Find \(\mathbf{a} \cdot \mathbf{b}\).
- Let vector \(\mathbf{v} = (-2, -1, 5)\) and vector \(\mathbf{w} = (2, 2, -4)\). Find \(\mathbf{v} \cdot \mathbf{w}\).
- Let vector \(\mathbf{p} = 2\mathbf{j} - 5\mathbf{k}\) and vector \(\mathbf{q} = -4\mathbf{i} + 3\mathbf{j} + 4\mathbf{k}\). Find \(\mathbf{p} \cdot \mathbf{q}\).
- Let vector \(\mathbf{u} = (-2, 4)\), vector \(\mathbf{v} = (-5, 5)\), and vector \(\mathbf{w} = (-1, 0)\). Evaluate \((\mathbf{u} \cdot \mathbf{v})\mathbf{w}\), if possible. If it is not possible, explain why not.
- Let vector \(\mathbf{a} = 3\mathbf{i} + 4\mathbf{j}\), vector \(\mathbf{b} = -2\mathbf{i} - 5\mathbf{j}\), and vector \(\mathbf{c} = \mathbf{i} - 5\mathbf{j}\). Evaluate \(\mathbf{a} \cdot \mathbf{b} \cdot \mathbf{c}\), if possible. If it is not possible, explain why not.
- Let vector \(\mathbf{r} = (3, 3, -1)\), vector \(\mathbf{s} = (0, 5, -3)\), and vector \(\mathbf{t} = (3, 0, 2)\). Evaluate \((\mathbf{r} \cdot \mathbf{s}) + \mathbf{t}\), if possible. If it is not possible, explain why not.
- Let vector \(\mathbf{x} = 2\mathbf{i} - \mathbf{k}\), vector \(\mathbf{y} = 2\mathbf{i} + 2\mathbf{j}\), and vector \(\mathbf{z} = -5\mathbf{i} - 5\mathbf{j} - 5\mathbf{k}\). Evaluate \((\mathbf{x} + \mathbf{y}) \cdot \mathbf{z}\), if possible. If it is not possible, explain why not.
- Let \(\theta \in [0, \pi]\) be the angle between two vectors \(\mathbf{u}\) and \(\mathbf{v}\). Recalling that \(\mathbf{u} \cdot \mathbf{v} = \| \mathbf{u}\| \| \mathbf{v}\| \cos \theta\), what can you say about \(\theta\) if \( \mathbf{u} \cdot \mathbf{v} = 0\)? What can you say about \(\theta\) if \( \mathbf{u} \cdot \mathbf{v} > 0\)? What can you say about \(\theta\) if \( \mathbf{u} \cdot \mathbf{v} < 0\)?
- Find the measure of the angle between vector \(\mathbf{u} = (-1, 1, 2)\) and vector \(\mathbf{v} = (-3, 1, -2)\).
- Find the measure of the angle between vector \(\mathbf{a} = \mathbf{i} + \mathbf{j}\) and vector \(\mathbf{b} = \mathbf{j} - \mathbf{k}\).
- Find the measure of the angle between vector \(\mathbf{v} = (1, -2, 1)\) and vector \(\mathbf{w} = (1, 1, -2)\).
- Find a non-zero vector \(\mathbf{p} \in \mathbb{R}^2\) that is orthogonal to \(\mathbf{q} = 2\mathbf{i} - 3\mathbf{j}\). Does there exist another non-zero vector \(\mathbf{r} \in \mathbb{R}^2\), not parallel to \(\mathbf{p}\), that is also orthogonal to \(\mathbf{q}\)?
- Find a non-zero vector \(\mathbf{x} \in \mathbb{R}^3\) that is orthogonal to \(\mathbf{y} = (4, 1, -2)\). Does there exist another non-zero vector \(\mathbf{z} \in \mathbb{R}^3\), not parallel to \(\mathbf{x}\), that is also orthogonal to \(\mathbf{y}\)?
- Let vector \(\mathbf{a} = 4\mathbf{i} + 5\mathbf{j}\) and vector \(\mathbf{b} = -\mathbf{i} - 2\mathbf{j}\). Find \(\text{proj}_{\mathbf{b}} \mathbf{a}\), the projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\).
- Let vector \(\mathbf{u} = (2, 2)\) and vector \(\mathbf{v} = (-5, 3)\). Decompose vector \(\mathbf{v}\) into the sum of a vector parallel to \(\mathbf{u}\) and a vector orthogonal to \(\mathbf{u}\).
- Let vector \(\mathbf{p} = -3\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) and vector \(\mathbf{q} = 3\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\). Decompose vector \(\mathbf{p}\) into the sum of a vector parallel to \(\mathbf{q}\) and a vector orthogonal to \(\mathbf{q}\).