Is scalar addition the same as vector addition?

Question

I need to know why also. Please help!!!

Answer 1

No. It's the same as n scalar additions, where n is the number of dimensions.

You can take a 2-d space, a plane, and describe it in terms of 2, 1-d coordinates, say, x and y. Then to find c = a + b, the resultant vector is

cx = ax +bx
cy = ay + by

Which is equivalent to TWO one dimensional (scalar) additions.

Answer 2

No.

Example:

Scalar addition. The current temp is 72F. You add 3 degrees. It is now 75F.
Essentially, it is normal addition.

Vector addition.
Two objects in motion collide. Because the objects have speed and direction, they can be considered vectors. In order to figure out the resultant of the collision you need to add the vector components (X and Y for lack of a better frame of reference) together.

HUH? What does that mean?

A vector has two components, magnitude and direction. Imagine you are standing at the origin of a large Cartesian plane. (x,y graph). You tie a string to a baseball, and throw it with all your might in the positive x,y direction. The string will fall on the floor making a straight line to where the ball landed. That is your vector. The length of string can be considered your magnitude, and the angle the string makes with the positive x axis can be considered your direction. (Throw the ball 50 feet at an angle of 30 degrees off the positive x axis.)

But, you could also describe the location of the ball using the x and y coordinates where the ball landed. How do you figure the x and y coordinates? Well, leaving the ball in place where it landed, you would walk out along the x axis and make a note where the perpendicular line to the ball strikes the x axis, then do the same thing along the y axis.
Or, you could use trigonometry.
What do you know? We know the hypotenuse and the angle. From that we can figure the sides. The x component of the vector is given through the magnitude of the vector times cosine of the vector angle, and the y is found through magnitude times sine of the angle. In this case:
50 cos 30 = 43.3 in the x direction
50 sin 30 = 25 in the y direction.

Now, do the same thing with another ball, but in a different direction. Lets say you throw the ball 25 feet in the 45 degree off the positive x axis direction.

To add the two vectors, you have to consider the x and the y components of each of the vectors. We already have the components of your first toss. Here are the second:
25 cos 45 = 17.677 in the x
25 sin 45 = 17.677 in the y

If you were to sum just the vectors without considering the x and y components, you would get:
50 at an angle of 30
25 at an angle of 45
------------------------------
75 at an angle of 75
which would be wrong. Experience and intuition would lead you to realize that resultant angle would be somewhere between the two tosses.

Instead, sum up the x and y components, then transform it back to vector notation.
x component = 43.3 + 17.677 = 60.977
y component = 25 + 17.677 = 42.677
and you would now place the summed ball at the grid coordinates of 60.977x and 42.677y

So how do we turn that back into a vector? You need to figure out the magnitude and the direction. For magnitude, Pythagoras is your friend: x squared + y squared = z squared. Or, for this problem, the hypotenuse (magnitude) is the square root of the sum of the squared sides.
60.677^2 + 423677^2 = z^2
3681.69 + 1821.38 = z^2
5503.08 = z^2
74.18 = z = vector magnitude.

Getting the direction is a little trickier, but again you can use trig to help. We know from our trig that the tangent of an angle is equal to the y component divided by the x component. So, we divide 42.677 by 60.977, then take the inverse tangent to get our angle.
34.99 degrees.

Sorry this was such a long explanation, I could have demonstrated this really easily if I could have drawn it. Hope this helps.

Answer 3

No. Scalar addition is the same as the addition of real numbers.

Answer 4

no, it's different because no-one cares which way scalars are pointing.