Magnitude of a Vector
The magnitude of a vector can be thought of as the size or the length of the vector. Given that a vector can be represented by an arrow on a coordinate plane then the length of that arrow would be the magnitude.
The length of the arrow can be found using the distance formula or more simply Pythagorean Theorem. Note that the distance formula and Pythagorean Theorem are essentially the same…
(1)The magnitude of the vector v is often given the symbol $|v|$.
In general a 3D vector, as shown below,
(2)has a magnitude given by the following equation:
This formula is given in the IB data booklet.
Negative Vectors
The two vectors
$v=\binom{3}{4}$ $w=\binom{-3}{-4}$
differ only in the +/- signs of the coordinates so that we can write $v=-w$ or $w=-v$. Graphing both vectors:
From the diagram, we can see that the vectors are in opposite directions. In general we can say that given two vectors $v$ and$-v$ the only difference between the two vectors is their direction and that $v$ points in the exact opposite direction from $-v$.
Vector Addition & Subtraction
Here's where vectors start to get a bit tricky…
Addition
Given the two vectors v and w below, we can ask the question what does it mean to add the two vectors, $v+w=?$ For argument sake lets use the two vectors below:
(4)Using the rules for matrix addition its not too much of a stretch to see that a new vector is formed by the addition of two vectors:
(5)The vector $\binom{5}{3}$ is the result of the addition and is called the resultant vector.
Geometrically this is a bit trickier to see. The vectors v and w are shown by red and blue lines respectively.
In order to add vectors geometrically one of the vectors (it doesn't matter which) is redrawn with its tail (non-pointy part) on the head (pointy part) of the second vector. (The redrawn vectors are shown by dashed lines.) Notice that the end point of the second vector is at the point 5,3ドル$ which matches the result from above. The green vector is the resultant vector.
The folks at University of Colorado Boulder (PhET) have a great simulation (shown below) - follow this link for a full screen version.
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Subtraction
If we can add vectors then we should be add to subtract them. Again if we look at the vectors v and w in their column form then subtraction is easy to understand
(6)Again the geometry of this can be a bit trickier. I personally can never remember how to do it, but there is a nice trick. We can say that:
(7)Now that may seem silly, but we have turned a subtraction problem into an addition problem where the vector $-w$ is simply in the opposite direction of $w$.
In the diagram above I have drawn the vector $-w$ and then simply moved it to the head of the vector $v$. Once again the resultant vector is shown in green and terminates at $(-3,1)$
Its also important to realize that $v-w \neq w-v$ just like 7ドル-4 \neq 4-7$. The two expressions are different by a +/- sign…
Multiplication by a Scalar
As seen above basic vector operations are easiest seen by returning to concepts of matrix operations. The same is true with Multiplication by a Scalar. Using the vector below as an example:
(8)If we multiply the vector by a scalar such as "2" then we have:
(9)Geometrically the effect of multiplication is simply to change the length of the vector. If the vector is multiplied by a scalar less than 1 the vector's length decreases.
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In the applet above the vector w is formed by multiplying v by a scalar:
(10)Initially the scalar has a value of 2. Drag the slider to change the value of the scalar. You can drag the tip of vector v to change the vector(s).