Dot Products and Vector Projections

In this article we will introduce a kind of vector multiplication in space-2 and space-3. The properties of the multiplication arithmetic will be determined and some of the applications will be given.

Suppose u and v are two nonzero vectors in space-2 and space-3, and suppose that these vectors have been located so that the starting point coincides. What we mean by the angle between u and v, is the angle Ɵ which is determined by u and v which satisfies 0Ɵ≤π, and is shown in the following image,


Definition
If u and v are vectors in space-2 or space-3 and Ɵ is the angle between u and v, then the product dot (dot product) or product in Euclidis (Euclidean Inner Product) u.v is defined as,

Example 1.
Determine the product of vector u = (0, 0, 1) and vector v = (0, 2, 2) with the angle between u and v is 450.
Solution:



Theorem 1.
Suppose u and v are vectors in Space 2 or space 3,
  1. v.v = |v|2; that is, |v|= (v.v)1/2
  2. if u and v are nonzero vectors and Ɵ is the angle between the two vectors, then

           Ɵ taper if and only if u.v> 0
           Ɵ blunt if and only if u.v <0
           Ɵ = π / 2 if and only if u.v = 0

Proof:
Because the angle Ɵ between v and v is 0, we get it
v.v = |v| |v| cos Ɵ = |v|2 cos Ɵ = |v|2

Because |u| > 0, |v| > 0, and u.v = |u| |v| cos Ɵ, then u.v has an equal sign like cos Ɵ. Because Ɵ satisfies Ɵ 0 π, then the angle Ɵ is acute if and only if cos Ɵ> 0; and Ɵ blunt if and only if cos Ɵ <0; and Ɵ = π / 2 if and only if cos Ɵ = 0.

Example 2.
If u = (1, -2, 3), v = (-3, 4, 2), and w = (3, 6, 3), then
u.v = (1)(-3) + (-2)(4) + (3)(2) = -5
v.w = (-3)(3) + (4)(6) + (2)(3) = 21
u.w = (1)(3) + (-2)(6) + (3)(3) = 0

So, u and v form blunt angles, v and w form sharp angles, and u and w are perpendicular to each other.

A perpendicular vector is also called an orthogonal vector. In the explanation of Theorem 1 (2), two non-zero vectors are perpendicular if and only if the result of the point is zero. If we agree that u and v are rather perpendicular then one or both of these vectors must be 0, because we can declare without exception that both vectors u and v will be orthogonal if and only if u.v is 0. To determine that u and v are orthogonal vector then we can write u _|_ v.

Row Vector Equation

Suppose i = (1, 0) and j = (0, 1) and note that these vectors are perpendicular and that the length is equal to one. These vectors i and j are called row vectors, because each vector u = (u1, u2)  can be expressed singly with i and j, namely:

u = (u1, u2) = u1(1, 0) + u2(0, 1) = u1i  + u2j

the geometric meaning of the relationship can be seen in Figure 3.

Example 3.
If u are arrows from P (2, 1) and Q (-3, 7), write u as u1i  + u2j
Solution:
We slide the arrow, so that its base coincides with the origin (Figure 4). This can be done by reducing the starting point component from the endpoint. Then obtained (-3-2, 7-(-1)=(-5, 8).
So that,

u = -5i + 8j

Example 4.
Determine the angle of ABC with A = (4, 3), B = (1, -1) and C = (6, -4).
Solution:

Based on Figure 5 we get,



Then the ABC angle is:


The following theorem will describe most of the important properties of the results of that point. The results of this point will be useful in calculations that include vectors.

Theorem 2.
If u, v, and w are vectors in space-2 or space-3 and k is scalar, then
  1. u.v = y.u commutative
  2. u.(v + w) = u.v + u.w (distributive)
  3. k(u.v) = (k u).v = u.(k v)
  4. v.v > 0 jika v ≠ 0 dan v.v = 0 jika v = 0

Proof:
We will prove (c) for vectors in Space-3 and leave the rest of the evidence as training for you.
Suppose u = (u1, u2, u3) and v = (v1, v2, v3)
Then,
k (u.v) = k ( u1 v1 + u2 v2 + u3 v3)
k (u.v) = (k u1) v1 + (k u2) v2 + (k u3) v3)
k (u.v) = (k u).v

As well,

K(u.v) = u. (k v)

In many applications this is interesting enough to "decipher" the vector u into the sum of two terms, the one equal to the vector a not zero while the other is perpendicular to a. If u and a are placed in such a way that the starting point will occupy point Q, we can describe the vector u as follows (Figure 6), decrease the perpendicular line from top u to the current through a, and form the vector w1 from Q to the line perpendicular to it. The next form will be different
w2 = u – w1


As shown in Figure 6, vector w1 is parallel to a, vector w2 is perpendicular to a, and
W1 + w2 = w1 + (u – w1) = u

We call the vector w1 an orthogonal projection u in a or sometimes we call the vector component u along a. We declare this
Proj a u

The vector w2 we call the vector component u is orthogonal to a. Because w2 = u - w1, we can write this vector in notation as
W2 = u – proj a u

The following theorem provides a formula for calculating proj a u and u – proj a u vectors

Theorem 3.
If u and a are vectors in space -2 or in space -3 and if a 0, then


proof:
Suppose that w1 = proj a u and w2 = u - proj a u. Because w1 is parallel to a, we must multiply a scalar, so we can write w1 = ka.
So,
u = w1 + w2 = ka + w2

By taking the results of the points from both sides with a, namely:
u . a = (ka + w2) .  a = k ||a||2 + w2 . a
But w2. a = 0 because w2 is perpendicular to a, so

Because proy a u = w1 = ka, we get it

Example 5. 
Suppose u = (2, -1, 3) and a = (4, -1, 2). Find the component vector u along a and the component vector u which is orthogonal to a.
Solution:
u . a = (2)(4) + (-1)(-1) + (3)(2) = 15
|a|2 = 42 + (-1)2 + 22 = 21

So, the component of vector u along a is


And the component vector u which is orthogonal to a is



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