Further problems 1

Q1

Draw two vectors $\vec{a}$ and $\vec{b}$ that are roughly similar to the ones shown in the figure below.

Determine graphically the vectors $-\vec{a}$, $\vec{a}+\vec{b}$, $\vec{a}-2\vec{b}$, and $-0.5 \vec{a}-\vec{b}$

Q2

Consider the two vectors $\vec a= \left(\begin{array}{r} 0\\ 1\\ 3 \end{array}\right)$ and $\vec b = \left(\begin{array}{r} -1\\ 2\\ 1 \end{array}\right)$

1. Draw the arrows described by these vectors. Assume they start at the origin.

2. Determine the components of the following vectors: $2\vec{a}$, $-1.5\vec{a}$, $3\vec{a}+5\vec{b}$, $-\vec{b}-\frac{1}{2}\vec{a}$, $3(-2\vec{b})$

3. Show with the help of a drawing that $\vert \vec{a}+\vec{b}\vert \leq \vert \vec{a}\vert + \vert\vec{b}\vert$. Verify using an example.

4. Determine the unit vector pointing in the same direction as $\vec{a}$.

5. Determine the unit vector pointing in the opposite direction of $\vec{a}$.

6. Find a vector of length 10 pointing in the same direction as $\vec{a}$.

7. Determine a vector of length 5 pointing in opposite direction of $\vec{b}$.

Q3

Determine the distance between the points $A(2\vert -4\vert 3)$ and $B(-1\vert 2\vert 1)$.

Q4

Consider the vector $\vec{u}=\left(\begin{array}{r} -1\\ 1\\ z \end{array}\right)$. Find all values $z$ such that $\vec{u}$ has magnitude $2$.

Q5

Consider the vector $\vec{u}=\left(\begin{array}{r} 0\\ 5\\ 3 \end{array}\right)$. Find another vector that is collinear to $\vec{u}$, and another one that is not.

Q6

Are the vectors $\vec{a}$ and $\vec{b}$ collinear?

1. $\vec{a}=\left(\begin{array}{r} 4\\ -1\\ 3 \end{array}\right)$ and $\vec{b}=\left(\begin{array}{r} -12\\ 3\\ -9 \end{array}\right)$
2. $\vec{a}=\left(\begin{array}{r} 3\\ -1\\ -0.1 \end{array}\right)$ and $\vec{b}=\left(\begin{array}{r} -2.25\\ 0.75\\ 0 \end{array}\right)$

Q7

Find components $x$ and $z$ such that the vectors $\vec{a}=\left(\begin{array}{r} -3\\ 1\\ 8 \end{array}\right)$ and $\vec{b}=\left(\begin{array}{r} x\\ -4\\ z \end{array}\right)$ are collinear.

Q8

Consider a straight line $g$ that passes through the points $A$ and $B$. Is the point $C$ on $g$?

1. $A(5\vert 2\vert 1)$, $B(10\vert -1\vert 0)$, $C(-8\vert 8\vert -6)$
2. $A(6\vert -3\vert 4)$, $B(2\vert 7\vert -5)$, $C(-4\vert 22\vert -18.5)$

Q9

Consider the points $A(6\vert 1\vert 0)$ and $B(10\vert 5\vert 4)$. Find the point $M$ on the segment between $A$ and $B$ such that it divides the segment with the ratio $2:1$.

Q10

Consider a triangle $ABC$, where the vertices are $A(6\vert 1\vert -3)$, $B(7\vert -7\vert 4)$, and $C(-4\vert 0\vert 5)$.

1. Find the circumference of the triangle.

2. Find a point $D$ such that the four points form a parallelogram. Is there more than one solution?

Q11

Consider the point $A(7\vert 1\vert 5)$. A point $B$ with x-coordinate $6$ and z-coordinate $-3$ has to be moved in y-direction such that the distance between $A$ and $B$ is exactly $9$. Determine the y-coordinate of $B$.

Q12

Find all the points on the z-axis such that the distance to the point $A(-6\vert 3\vert 7)$ is $7$.

Q13

A straight line passes through the points $A(4\vert 5\vert 3)$ and $B(2\vert 3\vert 1)$.

1. It intersects with the $xy$-plane at point $C$. Determine the coordinates of $C$.

2. It intersects with the yz-plane at point $C$. Determine the coordinates of $C$.

Q14

A straight line $g$ passes through the point $A(-2\vert 3\vert 4)$ and has direction $\vec{v}=\left(\begin{array}{r} 1\\ 2\\ -2 \end{array}\right)$ Find all points $U$ that are on line $g$ and have distance $4$ from point $A$.

Q15

Consider the cube shown below. Where does the straight line $g$ hit the $yz$-plane?

Solution

Solutions Check german version for update