Question Type 6: Finding the distance between two position vectors Exercises

Express the vector sum $\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}$ in $\mathbf{i},\mathbf{j},\mathbf{k}$ notation, given $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$ as in question 1.

In the $xy$-plane, vectors $\mathbf{p}=(5,2,0)$, $\mathbf{q}=(1,-3,0)$, $\mathbf{r}=(-4,1,0)$ and $\mathbf{s}=(0,0,0)$ are given. Use a single tip-to-tail chain to find the resultant and verify whether it equals the zero vector.

Given the vectors $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$, $\mathbf{c}=(1,1,1)$ and $\mathbf{d}=(0,2,2)$, calculate the resultant vector $\mathbf{R}=\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}$ without drawing individual tip-to-tail diagrams.

Three vectors in the $xy$-plane are $\mathbf{p}=(4,0,0)$, $\mathbf{q}=(0,3,0)$ and $\mathbf{r}=(-2,1,0)$. Without drawing separate tip-to-tail diagrams, find the resultant and its magnitude.

Given vectors $\mathbf{u}=2\mathbf{i}-\mathbf{j}+\mathbf{k}$, $\mathbf{v}=-\mathbf{i}+3\mathbf{j}+2\mathbf{k}$, $\mathbf{w}=\mathbf{i}+\mathbf{j}-\mathbf{k}$, find the missing vector $\mathbf{x}$ such that $\mathbf{u}+\mathbf{v}+\mathbf{w}+\mathbf{x}=\mathbf{0}$.

For the vectors $\mathbf{u}=(2,-1,0)$, $\mathbf{v}=(-1,2,3)$ and $\mathbf{w}=(0,1,-2)$, find $\mathbf{S}=\mathbf{u}+\mathbf{v}+\mathbf{w}$ and compute its magnitude $\|\mathbf{S}\|$.

Vectors $\mathbf{u},\mathbf{v},\mathbf{w},\mathbf{x}$ satisfy $\mathbf{u}+\mathbf{v}+\mathbf{w}=\mathbf{d}$ and $\mathbf{x}=\mathbf{d}-(\mathbf{u}+\mathbf{v}+\mathbf{w})$. If $\mathbf{u}=(1,2,3)$, $\mathbf{v}=(0,1,-1)$ and $\mathbf{d}=(3,3,4)$, find $\mathbf{w}$ and $\mathbf{x}$.

Compute the scalar projection of $\mathbf{R}=\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}$ onto $\mathbf{b}$, where $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$ are as in question 1.

Given $\mathbf{p}=(3,0,4)$, $\mathbf{q}=(0,5,-3)$ and $\mathbf{r}=(-2,1,1)$, find the unit vector in the direction of $\mathbf{p}+\mathbf{q}+\mathbf{r}$.

Given vectors $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$ from question 1, find the magnitude of $\mathbf{R}$ and compare with the sum of magnitudes $\|\mathbf{a}\|+\|\mathbf{b}\|+\|\mathbf{c}\|+\|\mathbf{d}\|$.

Show that the four vectors $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}$ from question 1 form a closed quadrilateral when placed tip-to-tail in order.

Let $\mathbf{a}=(1,1,3)$, $\mathbf{b}=(1,0,-1)$, $\mathbf{c}=(1,1,1)$ and $\mathbf{d}=(0,2,2)$. Determine the angle $\theta$ between $\mathbf{R}=\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}$ and $\mathbf{a}$.