Vectors and matrices › Vectors in the plane

Dot product

Solve the first problem 17 problems

These are 8 of the 17 problems. Sign in for free to see them all. Sign in

1. Weighted inner product of a sum ●○○○○ 10 ptsLet $\langle u,v\rangle = 4u_1v_1 + 5u_2v_2$ for vectors $u=(u_1,u_2)$ and $v=(v_1,v_2)$ in $\mathbb{R}^2$. Given $u=(3,-2)$, $v=(4,5)$ and $w=(-1,6)$, find $\langle u+v, w\rangle$.
2. Dot product with a linear combination ●○○○○ 10 ptsGiven that the angle between vectors $\vec{a}$ and $\vec{b}$ is $120^{\circ}$, and $|\vec{a}|=2$, $|\vec{b}|=5$, find $(2\vec{a}-\vec{b})\cdot\vec{a}$.
3. Triangle area from a dot product ●●○○○ 20 ptsIn triangle $ABC$, $AB=1$, $BC=3$, and $\overrightarrow{AB}\cdot\overrightarrow{BC}=-1$. Find the area of triangle $ABC$. Round to two decimal places.
4. Dot product from norms and cross product ●●○○○ 20 ptsLet $\mathbf{a}$ and $\mathbf{b}$ be vectors such that $\|\mathbf{a}\| = 2,$ $\|\mathbf{b}\| = 5,$ and $\|\mathbf{a} \times \mathbf{b}\| = 8.$ Find $|\mathbf{a} \cdot \mathbf{b}|.$
5. Dot product with a cross product of differences ●●○○○ 20 ptsLet $\mathbf{a} = \begin{pmatrix} -3 \\ 10 \\ 1 \end{pmatrix}$, $\mathbf{b} = \begin{pmatrix} 5 \\ \pi \\ 0 \end{pmatrix}$, and $\mathbf{c} = \begin{pmatrix} -2 \\ -2 \\ 7 \end{pmatrix}$. Compute $(\mathbf{a} - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a})]$.
6. Dot product with a cross product of unit vectors ●●●○○ 50 ptsLet $\mathbf{u}$ and $\mathbf{v}$ be unit vectors, and let $\mathbf{w}$ be a vector such that $\mathbf{u} \times \mathbf{v} + \mathbf{u} = \mathbf{w}$ and $\mathbf{w} \times \mathbf{u} = \mathbf{v}$. Compute $\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})$.
7. Dot product with circumcenter vector ●●●○○ 50 ptsIn triangle $ABC$, $AB = 3$ and $AC = 5$. Let $O$ be the circumcenter of triangle $ABC$. Find $\overrightarrow{OA} \cdot \overrightarrow{BC}$.
8. Side length from two dot products ●●●○○ 50 ptsIn $\triangle ABC$, $\overrightarrow{AB} \cdot \overrightarrow{AC} = \overrightarrow{AB} \cdot \overrightarrow{CB} = 4$. Find the length of side $AB$. Round to two decimal places.

More in Vectors in the plane

Components and additionLength of a vector combinationLinear combinations and collinear pointsMinimising dot productsParallel vectorsPerpendicular vectorsProjectionsVectors and triangle centres