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6. \(\mathbf{u} = 4\hat{\mathbf{i}} - \hat{\mathbf{j}} + 3\hat{\mathbf{k}}\) \(\mathbf{v} = 2\hat{\mathbf{i}} + \hat{\mathbf{j}} + \hat{\mathbf{k}}\) \(\mathbf{w} = -\hat{\mathbf{i}} + 5\hat{\mathbf{j}} + 3\hat{\mathbf{k}}\) \(\mathbf{r} = \alpha\hat{\mathbf{i}} + \beta\hat{\mathbf{j}} + \gamma\hat{\mathbf{k}}\ Evaluate the scalar products \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{v} \cdot \mathbf{w}\). Evaluate \(\mathbf{u} \cdot \mathbf{w}\) and interpret the result. If \(\mathbf{u}\) and \(\mathbf{v}\) are both perpendicular to \(\mathbf{r}\), find a unit vector in the direction of \(\mathbf{r}\) (without using the vector product). Find the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

          6.
\(\mathbf{u} = 4\hat{\mathbf{i}} - \hat{\mathbf{j}} + 3\hat{\mathbf{k}}\)
\(\mathbf{v} = 2\hat{\mathbf{i}} + \hat{\mathbf{j}} + \hat{\mathbf{k}}\)
\(\mathbf{w} = -\hat{\mathbf{i}} + 5\hat{\mathbf{j}} + 3\hat{\mathbf{k}}\)
\(\mathbf{r} = \alpha\hat{\mathbf{i}} + \beta\hat{\mathbf{j}} + \gamma\hat{\mathbf{k}}\
Evaluate the scalar products \(\mathbf{u} \cdot \mathbf{v}\) and \(\mathbf{v} \cdot \mathbf{w}\). Evaluate \(\mathbf{u} \cdot \mathbf{w}\) and interpret the result.
If \(\mathbf{u}\) and \(\mathbf{v}\) are both perpendicular to \(\mathbf{r}\), find a unit vector in the direction of \(\mathbf{r}\) (without
using the vector product).
Find the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

6.
𝐮 = 4𝐢̂ - 𝐣̂ + 3𝐤̂
𝐯 = 2𝐢̂ + 𝐣̂ + 𝐤̂
𝐰 = -𝐢̂ + 5𝐣̂ + 3𝐤̂
𝐫 = α𝐢̂ + β𝐣̂ + γ𝐤̂Evaluate the scalar products 𝐮·𝐯 and 𝐯·𝐰. Evaluate 𝐮·𝐰 and interpret the result.
If 𝐮 and 𝐯 are both perpendicular to 𝐫, find a unit vector in the direction of 𝐫 (without
using the vector product).
Find the angle between 𝐮 and 𝐯.

Added by Kristen M.

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