IB Mathematics Analysis and Approaches 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 $\mathbf{u}=\left(\begin{array}{c}3\\ 0\\ 4\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}4\\ 0\\ 1\end{array}\right)$.

Let $\mathbf{u}=\left(\begin{array}{c}4\\ -2\\ 1\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}1\\ 0\\ -1\end{array}\right)$.

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 $\mathbf{u}=\left(\begin{array}{c}-2\\ 5\\ 3\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}1\\ -4\\ 2\end{array}\right)$.

Consider the matrix $A$ and the column vector $w$ defined by: $$A=\left(\begin{array}{ccc}2& -1& 3\\ 0& 4& 1\\ -2& 5& 0\end{array}\right),w=\left(\begin{array}{c}1\\ 2\\ -1\end{array}\right)$$

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

Given $\mathbf{u}=\left(\begin{array}{c}2\\ 1\\ 0\end{array}\right)$ and $\mathbf{v}=\left(\begin{array}{c}-1\\ 3\\ 4\end{array}\right)$, 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 $\mathbf{u}(2)\cdot \mathbf{v}(2)$.