1.2 Vectors

Let $\va=\llt 2,0\rgt$ and $\vb=\llt 1,1\rgt$. Evaluate and sketch $\va+\vb,\ \va+2\vb$ and $2\va-\vb$.

Determine whether or not the given points are collinear (that is, lie on a common straight line)

Determine whether the given pair of vectors is perpendicular

Consider the vector $\va=\llt 3,4 \rgt$.

Consider the vector $\vb=\llt 3,4,0 \rgt$.

Let $\va=\llt a_1,a_2\rgt$. Compute the projection of $\va$ on $\hi$ and $\hj$.

Does the triangle with vertices $(1,2,3),\ (4,0,5)$ and $(3,6,4)$ have a right angle?

Show that the area of the parallelogram determined by the vectors $\va$ and $\vb$ is $|\va\times \vb|$.

Show that the volume of the parallelepiped determined by the vectors $\va,\ \vb$ and $\vc$ is

Verify by direct computation that

Consider the following statement: “If $\va\ne\vZero$ and if $\va\cdot\vb=\va\cdot\vc$ then $\vb=\vc$.” If the statment is true, prove it. If the statement is false, give a counterexample.

Consider the following statement: “The vector $\va\times(\vb\times\vc)$ is of the form $\al\vb+\be\vc$ for some real numbers $\al$ and $\be$.” If the statement is true, prove it. If the statement is false, give a counterexample.

What geometric conclusions can you draw from $\va\cdot(\vb\times\vc)=\llt 1,2,3\rgt$?

What geometric conclusions can you draw from $\va\cdot(\vb\times\vc)=0$?

Consider the three points $O=(0,0)$, $A=(a,0)$ and $B=(b,c)$.

Find the equation of a sphere if one of its diameters has end points $(2,1,4)$ and $(4,3,10)$.

Use vectors to prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Compute the areas of the parallelograms determined by the following vectors.

Consider the plane $W$, defined by:

Find the area of the parallelogram on $W$ defined by $0 \le x \le 3$, $0 \le y \le 2$.

Compute the volumes of the parallelepipeds determined by the following vectors.

Compute the dot product of the vectors $\va$ and $\vb$. Find the angle between them.

Determine the angle between the vectors $\va$ and $\vb$ if

Determine all values of $y$ for which the given vectors are perpendicular.

Let $\vu=-2\hi+5\hj$ and $\vv=\al\hi-2\hj$. Find $\al$ so that

Define $\va=\llt 1,2,3\rgt$ and $\vb=\llt 4,10,6\rgt$.

Compute $\llt 1,2,3\rgt\times\llt 4,5,6\rgt$.

Calculate the following cross products.

Let $\vp=\llt -1,4,2\rgt ,\ \vq=\llt 3,1,-1\rgt ,\ \vr=\llt 2,-3,-1\rgt$. Check, by direct computation, that

Calculate the area of the triangle with vertices $(0,0,0)$, $(1,2,3)$ and $(3,2,1)$.

A particle $P$ of unit mass whose position in space at time $t$ is $\vr(t)$ has angular momentum $L(t)=\vr(t)\times\vr'(t)$. If $\vr''(t)=\rho(t)\vr(t)$ for a scalar function $\rho$, show that $L$ is constant, i.e. does not change with time. Here $'$ denotes $\diff{}{t}$.

Show that the diagonals of a parallelogram bisect each other.

Consider a cube such that each side has length $s$. Name, in order, the four vertices on the bottom of the cube $A, B, C, D$ and the corresponding four vertices on the top of the cube $A', B', C', D'$.

Find the angle between the diagonal of a cube and the diagonal of one of its faces.

Consider a skier who is sliding without friction on the hill $y=h(x)$ in a two dimensional world. The skier is subject to two forces. One is gravity. The other acts perpendicularly to the hill. The second force automatically adjusts its magnitude so as to prevent the skier from burrowing into the hill. Suppose that the skier became airborne at some $(x_0,y_0)$ with $y_0=h(x_0)$. How fast was the skier going?

A marble is placed on the plane $ax+by+cz=d$. The coordinate system has been chosen so that the positive $z$–axis points straight up. The coefficient $c$ is nonzero and the coefficients $a$ and $b$ are not both zero. In which direction does the marble roll? Why were the conditions “$c\ne 0$” and “$a,b$ not both zero” imposed?

Show that $\va\cdot(\vb\times\vc) =(\va\times\vb)\cdot\vc$.

Show that $\va\times(\vb\times\vc) =(\va\cdot\vc)\vb-(\va\cdot\vb)\vc$.

Derive a formula for $(\va\times\vb)\cdot(\vc\times\vd)$ that involves dot but not cross products.

A prism has the six vertices

(Three dimensional Pythagorean Theorem) A solid body in space with exactly four vertices is called a tetrahedron. Let $A$, $B$, $C$ and $D$ be the areas of the four faces of a tetrahedron. Suppose that the three edges meeting at the vertex opposite the face of area $D$ are perpendicular to each other. Show that $D^2=A^2+B^2+C^2$.

(Three dimensional law of cosines) Let $A$, $B$, $C$ and $D$ be the areas of the four faces of a tetrahedron. Let $\al$ be the angle between the faces with areas $B$ and $C$, $\be$ be the angle between the faces with areas $A$ and $C$ and $\ga$ be the angle between the faces with areas $A$ and $B$. (By definition, the angle between two faces is the angle between the normal vectors to the faces.) Show that