Physical forces and velocities are not confined to the plane, therefore it is natural to extend the concept of vectors from two-dimensional space to three-dimensional space. In space, vectors are denoted by ordered triples v = <v1, v2, v3>, otherwise known as component form. The zero vector is denoted by 0 = < 0, 0, 0 >. Using the unit vectors i = < 1, 0, 0 > in the direction of the positive z-axis, the standard unit vector notation for v is v = v1i + v2j + v3k, otherwise known as unit vector form. If v is represented by the directed line segment from P(p1, p2, p3) to Q(q1, q2, q3), the component form of v is produced by subtracting the coordinates of the initial point from the coordinates of the terminal point, v ='<v1, v2, v3> = <q1-p1, q2-p2, q3-p3>.
Vectors in Space:
1. Two vectors are equal if and only if their corresponding components are equal.
2. The length of u = <u1, u2, u3> is ||u|| = (u1^2 + u2^2 + u3^2)^1/2 = vector length.
3. A unit vector u in the direction of v is given by u = v / ||v|| = unit vector.
4. The sum of u = <u1, u2, u3> and v = <v1, v2, v3> is u + v = <u1 + v1, u2 + v2, u3 + ||v3>.
5. The scalar multiple of the real number c and u = <u1, u2, u3> is cu = <cu1, cu2, cu3>.
6. The dot product of u = <u1, u2, u3> and v = <v1, v2, v3> is u x v = u1v1 + u2v2 + u3v3.
Example 1: Finding the Component Form of a Vector
Find the component form and length of the vector v having initial point (3,4,2) amd terminal point (3,6,4). Then find a unit vector in the direction of v.
The component form of v is:
v = <3-3, 6-4, 4-2> = <0,2,2>
This implies that the vector's length is
||v|| = (0^2 + 2^2 + 2^2)^1/2 = (8)^1/2 = 2(2)^1/2.
The unit vector in the direction of v is:
u = v / ||v|| = 1 / 2(2)^1/2 <0,2,2> = <0, 1/(2)^1/2, 1/(2)^1/2>.
Example 2: Finding the Dot Product of Two Vectors
Find the dot product of <0,3,-2> and <4,-2,3>.
<0,3,-2> x <4,-2,3> = 0(4) + 3(-2) + (-2)(3) = 0-6-6 = -12.
The angle between two nonzero vectors is the angle theta, 0 <(=) theta <(=) pi, between its respective standard position vectors. Thus, if theta is the angle between two nonzero vectors u and v, then cos(theta) =u x v / ||u|| ||v||. If the dot product of two nonzero vectors is zero, the angle between the vectors is 90 degrees. Such vectors are called orthogonal. For instance, the standard unit vectors i,j, and k are orthogonal to each other.
Example 3: Finding the Angle Between Two Vectors
Find the angle between u = <1,0,2> and v = <3,1,0>.
cos(theta) = u x v / ||u|| ||v|| = <1,0,2> x <3,1,0> / ||<1,0,2>|| ||<3,1,0>|| = 3 / (50)^1/2
This implies that the angle between the two vectors is
theta = arccos 3 / (50)^1/2 = 64.9 degrees.
Parallel Vectors:
In general two nonzero vectors u and v are parallel if there is some scalar c such that u = cv.
For example, the vectors shown below u, v, and w are parallel because u = 2v and w = -v.
Example 4: Parallel Vectors
Vector w has initial point (1,-2,0) and terminal point (3,2,1).
Which of the following vectors is parallel to w?
a) u = <4,8,2>
b) v = <4,8,4>
Begin by writing w in component form.
w = <3,-1, 2 -(-2), 1-0> = <2,4,1>
a) The vector u is parallel to w because
u = <4,8,2> = 2<2,4,1> = 2w
b) In this case, you need to find a scalar c such that <4,8,4> = c<2,4,1>.
However, equating corresponding components produces c = 2 for the first two components and c = 4 for the third. Hence, the equation has no solution, and the vectors are not parallel.
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