Vectors : Area of Parallelogram, Perpendicular Vectors , Angles Between Vectors, Orthogonal Vectors and Determinants

1. For vectors v and w in , show that v - w and v + w are perpendicular if and only if .

2. Let u = (-3, 1, 2), v = (4, 0, -8), and w = (6, -1, -4) be vectors in . Find the components of the vector x that satisfies 2u - v + x = 7x + w.

3. Find a non-zero u vector such that satisfies the following.
a. u has the same direction as v = (4, -2, -1) and has initial point P(-1, 3, -5).
b. u has the opposite direction of v = (4, -2, -1) and has initial point P(-1, 3, -5).
c. u is perpendicular to v = (4, -2, -1), and has a magnitude of 2.
d. Let u = (2, 1, 1) and v = (4, -2, -1). Find a vector w which is perpendicular to both u and v. Be sure to prove you have found such a vector.

4. Define a function on vectors from which describes the distance between two vectors as,
, where (the square root of the dot product of x and itself)
Show that this distance function has the following 4 properties.
a. , for all vectors u and v.
b. , if and only if u = v.
c.
d. (Hint: us part d of theorem 4.1.4)

5. true or false. For those which are true, give a reason, and for those which are false, give a counterexample.
a. The cross-product u x v is perpendicular to both u and v.
b. The determinant of a 2x2 matrix is a vector.
c. The determinant of a 3x3 matrix is zero if two rows of the matrix are parallel vectors in .

d. In order for the determinant of a 3x3 matrix to be zero, it must be true that two rows of the matrix are parallel vectors in .

e. The area of a parallelogram in determined by non-zero vectors u and v is given by the formula where A is the matrix whose rows are u and v. ie

f. The area of a parallelogram in determined by non-zero vectors u and v is given by the formula , where .

g. If the angle between vectors u and v is , then

h. For any vector u in , we have

i. for all vectors in

Please see the attached file for the fully formatted problems.

j. for all orthogonal vectors in
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