Scalar multiplication of vectors:
When you multiply a vector by a scalar, the direction of the vector is unchanged; only the magnitude changes. Let
A=<a,b> and c be some real number. Then c*
A=<a*c,b*c>.
Q: Why doesn't the direction change?
A: Think of it this way, if you have been in algebra, you should know that the slope of a line is the rise(change in y) divided by the run (change in x). Well, that is just what b and a are, respectively. So the slope of that vector is the direction of that vector. When you multiply both a and b by c, nothing changes in the slope, because c "cancels out" in the division.
Addition of vectors:
Q: How do you add/subtract vectors?
A: It is easy, once in Cartesian coordinates. Let
A=<a,b> and
B=<c,d>. Then
A+
B=<a+c,b+d>. If you wanted to subtract
B instead of add it, you could just multiply it by -1, as above, then add just like you would normally.
The Scalar Product (The Dot Product)
Given two vectors,
A and
B, the scalar product of the two, in mathematical terms is written
A·
B=|
A|*|
B|*cos(theta), where theta is acute (or 180) angle between them. What does || mean, though? Well || returns the magnitude of the vector, making a scalar quantity. Since the dot product only involves the magnitudes of vectors, the dot prodect itself is a scalar quantity, hence the name scalar product. Using a previous example, speed is the magnitude of velocity.
Q: How do you find the magnitude of a vector?
A: Let
v=<a,b>. Then, |
v|=sqrt(a^2+b^2). For those with a good memory, you might recognize that to be just like the formula for finding r from Cartesian coordinates. That is because r is the magnitude of the vector.
Q: What if you don't know the angle between them?
A: That is one of the uses of the dot product. There is a shortcut way of figuring out the answer of the dot product itself. Let
A=<a,b> and
B=<c,d>. Then,
A·
B=a*c+b*d. And you already know how to find the magnitude of
A and
B. So just divide both sides by those magnitudes, and you are left with (
A·
B)/(|
A|*|
B|)=cos(theta). With the help of a calculator, you can find that angle.
Next post: Vector product (cross product)