Question Type 1: Finding the dot product between two vectors using different operations Exercises

Determine the value of $\lambda$ such that the vectors $\mathbf{u}=(1, \lambda, 3)$ and $\mathbf{v}=(2, -1, 4)$ are perpendicular.

Find the scalar product of the vectors $\mathbf{u} = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 4 \\ 0 \\ 1 \end{pmatrix}$.

Let $\mathbf{u} = \begin{pmatrix} 4 \\ -2 \\ 1 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix}$.

Find the scalar projection of $\mathbf{u}$ onto $\mathbf{v}$.

Compute the dot product of the four-dimensional vectors $\mathbf{u}=(1,2,3,4)$ and $\mathbf{v}=(0,-1,2,-3)$.

Find the dot product of the five-dimensional vectors $\mathbf{u}=(1,0,-2,3,1)$ and $\mathbf{v}=(2,1,0,-1,4)$.

Given two vectors $\mathbf{u}$ and $\mathbf{v}$ satisfy $|\mathbf{u}|=5$, $|\mathbf{v}|=7$ and the angle between them is $60^\circ$, find $\mathbf{u}\cdot \mathbf{v}$.

Find the scalar product of $\mathbf{u}=\begin{pmatrix}-2\\5\\3\end{pmatrix}$ and $\mathbf{v}=\begin{pmatrix}1\\-4\\2\end{pmatrix}$.

Consider the matrix $A$ and the column vector $w$ defined by: $A=\begin{pmatrix}2 & -1 & 3\\0 & 4 & 1\\-2 & 5 & 0\end{pmatrix}, \quad w=\begin{pmatrix}1\\2\\-1\end{pmatrix}$

Calculate the dot product of the first row of $A$ and the vector $w$.

Given $\mathbf{u}=\begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}$ and $\mathbf{v}=\begin{pmatrix} -1 \\ 3 \\ 4 \end{pmatrix}$, calculate the scalar product $(\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}-\mathbf{v})$.

Let $\mathbf{u}=(\cos\theta,\sin\theta)$ and $\mathbf{v}=(1,1)$. Express $\mathbf{u}\cdot\mathbf{v}$ in terms of $\theta$.

Let $\mathbf{u}(t)=(t,t^2,1)$ and $\mathbf{v}(t)=(1,2t,t)$. Compute $\mathbf{u}(2)\cdot \mathbf{v}(2)$.