MATH 120/125Linear Algebra M.Solomonovich

Assignment # 2 SOLUTIONSVector algebra: dot product; cross product; triple product.

  1. It is known that Find all the values of t, for which vectors 3a + tb and 3a – tb are orthogonal.

Solution.

Let us write the orthogonality condition for the vectors 3a + tb and 3a – tb:

  1. It is given:. Find the norm of the vector m + 3n.

Solution.

For any vector v, . Then we can find the norm of m + 3n:

  1. a, b, and c are three unit vectors satisfying the condition . Evaluate .

Let us multiply the given condition by a; we shall obtain . Similarly, by multiplying the given condition by b and c we can obtain: . Let us add the obtained three equalities: , which, taking into account that (since a, b, and c are unit vectors) and the commutative property of dot product, can be rewritten as .

It is interesting to observe the geometric meaning of this result.

First, let us notice that for a unit vector u, its projection onto a vector v is equal to the cosine of the angle between v and u: .

Also, a projection of a vector onto a unit vector is equal to their dot product (why?).

Vectors a, b, and c form a regular triangle with the sides of length 1(see the figure below). Then, =.

  1. Vectors a and b are given by their components in the standard basis: a = (1, -2, 2); b = (1, 0, 1). Find: (i) the orthogonal projection of a onto a – 2b; (ii) an angle between a and a + b.

Solution.

; Then .

(It is not specified whether the scalar or vector projection to be found; the vector projection is ).

We can determine the cosine of the angle: .

  1. Evaluate the area of a parallelogram formed by the vectors 3a + b and a – b, if .

Solution.

  1. Vectors a and b are given by their components in the standard basis :

a = (2, -1, 1); b = (1, 0, -1). Find the unit vector perpendicular to both vectors a and b and forming an obtuse angle with the positive Y-semi-axis.

Solution.

A vector v perpendicular to a and b will be a scalar multiple of their cross product.

. Then a unit vector collinear with v will be .

Since the required vector forms an obtuse angle with the Y-axis, its second component must be negative (the cosine of an obtuse angle is negative); hence the required vector is .

  1. Four points are given by their coordinates: A(1, 0, 2); B(0, 1, 3); C(1, 1, -1); and D(1, 1, 4) .

(i)Evaluate the area of triangle BCD;

(ii)Determine, if these four points are lying in one plane;

(iii)Evaluate the volume of the pyramid ABCD.

Solution.

(i)The area is a half of the area of a parallelogram formed by any two vectors that represent sides of the triangle: .

(ii)Four points are coplanar iff any four vectors joining all of them are coplanar. The latter are coplanar iff their triple product is zero. ; hence the points are not coplanar.

(iii)