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# Give a Cartesian for planes through given points spanned by given vectors

Consider the vectors

1. Find a nonzero vector perpendicular to both and .
2. Find a Cartesian equation for the plane through which is spanned by and .
3. Find a Cartesian equation for the plane through which is spanned by and .

1. Since and are independent, we can take

2. From part (a) we have is perpendicular to both and , so a Cartesian equation for the plane is given by

Further, since the point is on the plane, we must have . Hence, the Cartesian equation for the plane is

3. Again, we have a Cartesian equation for the plane given by

Since is on the plane we must have

Hence, the Cartesian equation for the plane is given by

# Using Cramer’s rule, solve the given system of equations

Use Cramer’s rule to solve the system of equations:

From Cramer’s rule we have,

# Solve a system of equations using Cramer’s rule

Use Cramer’s rule to solve the system of equations:

From Cramer’s rule we have,

# Solve a given system of equations using Cramer’s rule

Use Cramer’s rule to solve the system of equations:

From Cramer’s rule we have

# Give a vector based proof of Heron’s formula for computing the area of a triangle

Let denote the area of a triangle with sides of lengths . Heron’s formula states that

We prove this formula using vectors via the following steps. Assume the triangle has vertices , and with

1. Using the identities

prove the formula

2. Simplify the formula in part (a) to obtain the formula

and use this to deduce Heron’s formula.

1. Proof. We know the area of the triangle (in terms of the vectors and ) is

Therefore,

2. Proof. We simplify the formula in part (a),

# Prove a formula for the perpendicular distance between a point and a line

1. Assume that . Prove that the perpendicular distance from to the line passing through the points and is given by the formula

2. Compute the distance in the case

1. Proof. We know the area of the parallelogram determined by and is given by

But this is exactly twice the area of the triangle with base and height the perpendicular line from to the line through and . Hence, is the perpendicular distance from to the line through and , and

2. When we have

# Prove a vector formula for the volume of a tetrahedron

1. Consider a tetrahedron with vertices . Prove that the volume of the tetrahedron is given by the formula

2. Compute the volume in the case that

1. Proof. We know the volume of a tetrahedron is given by (where denotes the altitude of the tetrahedron). We know (page 490 of Apostol) that the volume of the parallelepiped with base formed by vector and height formed by vector is given by . In this case we have that the base of the tetrahedron is formed by the vectors and , and the height is formed by the vector . Further, we know that the area of the base described by the vectors and is one half that of the parallelepiped whose base is given by vectors and (since the base of the parallelepiped described by vectors and is a rectangle, and the base of the tetrahedron is the triangle formed by cutting this rectangle along the diagonal). Therefore we have

2. Using the formula in part (a) with the given values of we have

# Prove or disprove a given vector formula

Prove or disprove:

Proof. We can compute using the identities we have derived in the previous exercises of this section,

# Prove an identity relating scalar triple products of vectors A,B,C

Prove the following identity holds for vectors .

Proof. From a previous exercise (Section 13.14, Exercise #9(d)), we know

So, with in place of and in place of in this formula we have

# Compute the cross product of given vectors in terms of the unit coordinate vectors

Let such that

In terms of the unit coordinate vectors compute the cross product

Using part (a) of the previous exercise and equation (13.10) on page 490 of Apostol () we compute

Then we use the other given relations

Which all implies