APPLY SCALAR (DOT) PRODUCT OF TWO VECTORS

DEFINITION OF SCALAR PRODUCT

The dot product, also called the scalar product, of two vector s is a number ( scalarquantity) obtained by performing a specific operation on the vector components. The dot product has meaning only for pairs of vectors having the same number of dimensions. The symbol for dot product is a heavy dot ( ).

In the two-dimensional Cartesian plane, vectors are expressed in terms of the x -coordinates and y -coordinates of their end points, assuming they begin at the origin ( x , y ) = (0,0). Some examples are shown in the illustration below.

The dot product of two vectors is determined by multiplying their x -coordinates, then multiplying their y -coordinates, and finally adding the two products. Thus, in the above example:

A  B = (2 x -4) + (5 x -3) = -8 - 15 = -23

B  C = (-4 x 5) + (-3 x -5) = -20 + 15 = -5

C  A = (5 x 2) + (-5 x 5) = 10 - 25 = -15

In polar coordinates, vectors are expressed in terms of length (magnitude) and direction. When expressed in this format, the dot product of two vectors is equal to the product of their lengths, multiplied by the cosine of the angle between them.

For any two vectors A and B , A  B = B  A . That is, the dot product operation is commutative; it does not matter in which order the operation is performed.

Properties of Dot Product

Another property of the dot product is:

(au + bv) · w = (au) · w + (bv) · w, where a and b are scalars

Here is the list of properties of the dot product:

1. u · v = |u||v| cos θ
2. u · v = v · u
3. u · v = 0 when u and v are orthogonal.
4. 0 · 0 = 0
5. |v|² = v · v
6. a (u·v) = (a u) · v
7. (au + bv) · w = (au) · w + (bv) · w

Calculating

The Dot Product gives a number as an answer (a "scalar", not a vector).

The Dot Product is written using a central dot:

a · b
This means the Dot Product of a and b

We can calculate the Dot Product of two vectors this way:

Where:
|a| is the magnitude (length) of vector a
|b| is the magnitude (length) of vector bθ is the angle between a and b

So we multiply the length of a times the length of b, then multiply by the cosine of the angle between a and b

OR we can calculate it this way:

a · b = a_x × b_x + a_y × b_y

So we multiply the x's, multiply the y's, then add.

    

Both methods work!

😁😁😁😁😁😁😁😁😁

Example: Calculate the dot product of vectors a and b:

a · b = |a| × |b| × cos(θ)

a · b = 10 × 13 × cos(59.5°)

a · b = 10 × 13 × 0.5075...

a · b = 65.98... = 66 (rounded)

or we can calculate it this way:

a · b = a_x × b_x + a_y × b_y

a · b = -6 × 5 + 8 × 12

a · b = -30 + 96

a · b = 66

Both methods came up with the same result (after rounding)

Also note that we used minus 6 for a_x (it is heading in the negative x-direction)

Note: you can use the Vector Calculator to help you.