Monday, March 19, 2007

Section 10-2: Two Dimensional Vector Practice

Hey kids in this lesson we reviewed our vectors. We learned a bunch of new vocab which I will give you further down. Hope you don't find this too frustrating. Also, I couldn't figure out how to get the arrows above the letters, so if you think there should be an arrow above it, it's because there should be.

Vector Definitions and Properties
Vector Quantity - has both magnitude and direction (eg. - force, velocity, displacement)
Scalar - quantity but no direction (eg. - weight, height)
Vector - directed line segment that represents a vector quantity (v w/ an arrow above it)
Tail - where the vector begins
Head - where the vector ends
Magnitude - (absolute value is used) the length of a vector
absolute value(v), If v = xi = xj, Then absolute value(v) = square root (x² + y²)
Unit Vector - absolute value(u) in the direction of v is a vector that is one unit long in the same direction as v. So u = v/abs(v)* - divide the vector by it's length
Vectors are equal if they have the same magnitude and direction.
The opposite of a vector is the same length in the opposite direction (-v)
Position Vector - v = xi + yj; Starts at the origin and the end point is (x,y)
When adding two vectors the resultant vector falls from the tail of the 1st vector to the tip of the 2nd. (Start of the 1st to the end of the 2nd)
* abs() = absolute value

Example Problem 1
If a = 4i + 8j and b = 5i - 3j
Find a+b => Add the two equations => 9i -5j
Find a-b => Subtract the two equations=> -i + 11j
Find -a => -4i - 8j
Find 2a + 4b => First multiply a by 5 and b by 4 => 8i +16j and 20i - 12 j => then add => 28i +4j
When doing problems like this make sure you understand that the abs(a) + abs(b) does NOT equal the abs(a + b)
Also, when finding the angle for the vector, use the inverse tangent.

Example Problem 2
Given point A (5, 10) and B (8, 20)
Find vector AB (Pointing from A to B)
to do this, look at the given points and subtract. AB = 3i - 10j
Find the position vector of the point 3/4 of the way from A to B.
To do this, it is easier to draw the postion vector to point A, then determine what is 3/4 of AB. Once that is found, add the two vectors.
A = 5i + 10j; 3/4 AB = 3/4(3i - 10j) = (9/4)i - (30/4)j
The answer would then be 5i+10j + (9/4)i - (30/4)j
This equals => 7.25i + 2.5j

Now in honor of March Madness, and USC beating Texas, here is a picture of the Trojan basketball team.


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