IB Mathematics Analysis and Approaches Question Type 3: Doing Simple Calculations with Vectors Exercises

Compute $3v+w$ where $v=\left(\begin{array}{c}4\\ 6\\ 1\end{array}\right)$ and $w=\left(\begin{array}{c}3\\ 4\\ 3\end{array}\right)$.

Given $\mathbf{v}=(4,6,1)$ and $\mathbf{w}=(3,4,3)$, compute $-\mathbf{v}+4\mathbf{w}$.

Determine a unit vector perpendicular to both $\mathbf{v}=\left(\begin{array}{c}4\\ 6\\ 1\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}3\\ 4\\ 3\end{array}\right)$.

Compute the distance between the points represented by $v=(4,6,1)$ and $2w=2(3,4,3)$.

Determine the magnitude of the vector $-\mathbf{v}+4\mathbf{w}$ given $\mathbf{v}=(4,6,1)$ and $\mathbf{w}=(3,4,3)$.

Calculate the dot product $\mathbf{v}\cdot \mathbf{w}$ given $\mathbf{v}=\left(\begin{array}{c}4\\ 6\\ 1\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}3\\ 4\\ 3\end{array}\right)$.

Find the projection of $\mathbf{v}=(4,6,1)$ onto $\mathbf{w}=(3,4,3)$.

Compute $2\mathbf{v}-3\mathbf{w}$ for $\mathbf{v}=\left(\begin{array}{c}4\\ 6\\ 1\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}3\\ 4\\ 3\end{array}\right)$.

Calculate the angle $\theta$ between the vectors $\mathbf{v}=\left(\begin{array}{c}4\\ 6\\ 1\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}3\\ 4\\ 3\end{array}\right)$.

Find the cross product $\mathbf{v}\times \mathbf{w}$ for $\mathbf{v}=\left(\begin{array}{c}4\\ 6\\ 1\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}3\\ 4\\ 3\end{array}\right)$.

Find $\mathbf{v}-2\mathbf{w}$ for $\mathbf{v}=(4,6,1)$ and $\mathbf{w}=(3,4,3)$.

Find the unit vector in the direction of $-\mathbf{v}+4\mathbf{w}$ for $\mathbf{v}=\left(\begin{array}{c}4\\ 6\\ 1\end{array}\right)$ and $\mathbf{w}=\left(\begin{array}{c}3\\ 4\\ 3\end{array}\right)$.