VectorsVectors
`
`This is a vector:
`
`Advanced
`
`A vector has magnitude (size) and direction:
`
`The length of the line shows its magnitude and the arrowhead points in the direction.
`
`We can add two vectors by joining them head-to-tail:
`
`And it doesn't matter which order we add them, we get the same result:
`
`Example: A plane is flying along, pointing North, but there is a wind
`coming from the North-West.
`
`Advanced
`
`Legend3D, Inc.
`Exhibit 1024-0001
`
`
`
`This is a vector:
`
`Advanced
`
`A vector has magnitude (size) and direction:
`
`The length of the line shows its magnitude and the arrowhead points in the direction.
`
`We can add two vectors by joining them head-to-tail:
`
`And it doesn't matter which order we add them, we get the same result:
`
`Example: A plane is flying along, pointing North, but there is a wind
`coming from the North-West.
`
`Advanced
`
`Legend3D, Inc.
`Exhibit 1024-0001
`
`
`
`The two vectors (the velocity caused by the propeller, and the velocity of the wind)
`result in a slightly slower ground speed heading a little East of North.
`
`If you watched the plane from the ground it would seem to be slipping sideways a
`little.
`
`Have you ever seen that happen? Maybe you have seen birds struggling against a
`strong wind that seem to fly sideways. Vectors help explain that.
`
`Velocity , acceleration , force and many other things are vectors.
`
`Subtracting
`
`We can also subtract one vector from another:
`
`first we reverse the direction of the vector we want to subtract,
`then add them as usual:
`
`Legend3D, Inc.
`Exhibit 1024-0002
`
`
`
`a − b
`
`Notation
`
`A vector is often written in bold, like a or b.
`
`A vector can also be written as the letters
`of its head and tail with an arrow above it, like this:
`
`
`
`Calculations
`
`Now ... how do we do the calculations?
`
`The most common way is to first break up vectors into x and y parts, like this:
`
`The vector a is broken up into
`the two vectors ax and ay
`
`(We see later how to do this.)
`
`Adding Vectors
`
`We can then add vectors by adding the x parts and adding the y parts:
`
`Legend3D, Inc.
`Exhibit 1024-0003
`
`
`
`The vector (8,13) and the vector (26,7) add up to the vector (34,20)
`
`Example: add the vectors a = (8,13) and b = (26,7)
`
`c = a + b
`
`c = (8,13) + (26,7) = (8+26,13+7) = (34,20)
`
`Subtracting Vectors
`
`To subtract, first reverse the vector we want to subtract, then add.
`
`Example: subtract k = (4,5) from v = (12,2)
`
`a = v + −k
`
`a = (12,2) + −(4,5) = (12,2) + (−4,−5) = (12−4,2−5) = (8,−3)
`
`Magnitude of a Vector
`
`The magnitude of a vector is shown by two vertical bars on either side of the vector:
`
`|a|
`
`OR it can be written with double vertical bars (so as not to confuse it with absolute value):
`
`We use Pythagoras' theorem to calculate it:
`
`||a||
`
`|a| = √( x2 + y2 )
`
`Legend3D, Inc.
`Exhibit 1024-0004
`
`
`
`Example: what is the magnitude of the vector b = (6,8) ?
`
`|b| = √( 62 + 82 ) = √( 36+64 ) = √100 = 10
`
`A vector with magnitude 1 is called a Unit Vector .
`
`Vector vs Scalar
`
`A scalar has magnitude (size) only.
`
`Scalar: just a number (like 7 or −0.32) ... definitely not a vector.
`
`A vector has magnitude and direction, and is often written in bold, so we know it is not
`a scalar:
`
`so c is a vector, it has magnitude and direction
`but c is just a value, like 3 or 12.4
`
`Example: kb is actually the scalar k times the vector b.
`
`Multiplying a Vector by a Scalar
`
`When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big
`or small the vector is.
`
`Example: multiply the vector m = (7,3) by the scalar 3
`
` a = 3m = (3×7,3×3) = (21,9)
`
`Legend3D, Inc.
`Exhibit 1024-0005
`
`
`
`It still points in the same direction, but is 3 times longer
`
`(And now you know why numbers are called "scalars", because they "scale" the vector up or
`down.)
`
`
`
`Multiplying a Vector by a Vector (Dot Product and Cross Product)
`
`How do we multiply two vectors together? There is more than one
`way!
`
`The scalar or Dot Product (the result is a scalar).
`The vector or Cross Product (the result is a vector).
`
`(Read those pages for more details.)
`
`
`
`More Than 2 Dimensions
`
`Vectors also work perfectly well in 3 or more dimensions:
`y
`
`(1, 4, 5)
`
`z
`
`The vector (1,4,5)
`
`x
`
`Legend3D, Inc.
`Exhibit 1024-0006
`
`
`
`Example: add the vectors a = (3,7,4) and b = (2,9,11)
`
`c = a + b
`
`c = (3,7,4) + (2,9,11) = (3+2,7+9,4+11) = (5,16,15)
`
`Example: what is the magnitude of the vector w = (1,−2,3) ?
`
`|w| = √( 12 + (−2)2 + 32 ) = √( 1+4+9 ) = √14
`
`Here is an example with 4 dimensions (but it is hard to draw it!):
`
`Example: subtract (1,2,3,4) from (3,3,3,3)
`
`(3,3,3,3) + −(1,2,3,4)
`= (3,3,3,3) + (−1,−2,−3,−4)
`= (3−1,3−2,3−3,3−4)
`= (2,1,0,−1)
`
`
`
`Magnitude and Direction
`
`We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):
`
`<=>
`
`Vector a in Polar
`Coordinates
`
`
`
`Vector a in Cartesian
`Coordinates
`
`You can read how to convert them at Polar and Cartesian Coordinates , but here is a quick
`summary:
`
`Legend3D, Inc.
`Exhibit 1024-0007
`
`
`
`From Polar Coordinates (r,θ)
`to Cartesian Coordinates (x,y)
`
`x = r × cos( θ )
`y = r × sin( θ )
`
` From Cartesian Coordinates (x,y)
`to Polar Coordinates (r,θ)
`r = √ ( x2 + y2 )
`θ = tan-1 ( y / x )
`
`
`
`
`
`An Example
`
`Sam and Alex are pulling a box.
`
`Sam pulls with 200 Newtons of force at 60°
`Alex pulls with 120 Newtons of force at 45° as shown
`
`What is the combined force , and its direction?
`
`Let us add the two vectors head to tail:
`
`First convert from polar to Cartesian (to 2 decimals):
`
`Sam's Vector:
`
`x = r × cos( θ ) = 200 × cos(60°) = 200 × 0.5 = 100
`y = r × sin( θ ) = 200 × sin(60°) = 200 × 0.8660 = 173.21
`
`Alex's Vector:
`
`x = r × cos( θ ) = 120 × cos(-45°) = 120 × 0.7071 = 84.85
`y = r × sin( θ ) = 120 × sin(-45°) = 120 × -0.7071 = −84.85
`
`Now we have:
`
`Legend3D, Inc.
`Exhibit 1024-0008
`
`
`
`It is now easy to add them:
`
`(100, 173.21) + (84.85, −84.85) = (184.85, 88.36)
`
`That answer is valid ... but let's convert back to polar, as the question was in polar:
`
`r = √ ( x2 + y2 ) = √ ( 184.852 + 88.362 ) = 204.88
`θ = tan-1 ( y / x ) = tan-1 ( 88.36 / 184.85 ) = 25.5°
`
`And we have this (rounded) result:
`
`And it looks like this for Sam and Alex:
`
`They might get a better result if they were shoulder-to-shoulder!
`
`
`
`Legend3D, Inc.
`Exhibit 1024-0009
`
`
`
`Question 1 Question 2 Question 3 Question 4 Question 5
`Question 6 Question 7 Question 8 Question 9 Question 10
`
`Search :: Index :: About :: Contact :: Contribute :: Cite This Page :: Privacy
`
`Copyright © 2016 MathsIsFun.com
`
`Legend3D, Inc.
`Exhibit 1024-0010
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