Question Type 1: Finding the dot product of two different vectors Exercises

Two unit vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $\mathbf{u} \cdot \mathbf{v} = \frac{1}{2}$. Find the angle between $\mathbf{u}$ and $\mathbf{v}$.

In a parallelogram, the diagonals are given by $\mathbf{a} + \mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ for $\mathbf{a} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} 1 \\ 3 \\ 4 \end{pmatrix}$. Find the angle between the diagonals.

Determine the value of $k$ such that the vectors $(k, 2, 3)$ and $(1, k, 4)$ are perpendicular.

Determine whether the vectors $\mathbf{a} = (1, 0, 1)$ and $\mathbf{c} = (5, 6, 1)$ are perpendicular.

Show that the angle between the vectors $(1, 1, 1)$ and $(1, 0, 0)$ is acute and find its measure.

Given the vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 5 \\ 6 \\ 1 \end{pmatrix}$, find the value of $\mathbf{b} \cdot (\mathbf{a} - \mathbf{c})$.

Calculate the value of $\mathbf{b} \cdot \mathbf{c} - 2\mathbf{a} \cdot \mathbf{c}$ for $\mathbf{a} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 5 \\ 6 \\ 1 \end{pmatrix}$.

Verify the distributive property $\mathbf{a} \cdot (\mathbf{b} + 2\mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + 2\mathbf{a} \cdot \mathbf{c}$ for $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}$ and compute the resulting value.

Find the angle between the vectors $\mathbf{p}=(3,-1,2)$ and $\mathbf{q}=(1,2,-2)$.

Find the angle between the vectors $\mathbf{u} = \begin{pmatrix} 1 \\ 4 \\ 3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ 1 \\ 2 \end{pmatrix}$.

Compute the dot product $\mathbf{a} \cdot (\mathbf{b} + 2\mathbf{c})$ for $\mathbf{a} = (1, 0, 1)$, $\mathbf{b} = (3, 2, 1)$ and $\mathbf{c} = (5, 6, 1)$.

Find all real values of $t$ such that the vectors $(1, t, t^2)$ and $(2, -1, 1)$ are orthogonal.