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Points, Vectors, and Functions

Points, Vectors, and Functions


With an understanding that vectors are simply mathematical equivalents of mythological hybrid beasts, we will investigate their parts to see if we can understand them better. Surely, with the help of the minotaur and griffin vectors, the calculus bear stands no chance against you, right?

The magnitude of a vector is like the body of the vector. We know how much vector is there. We use the notation
|||| for the magnitude of the vector . This is like the notation for absolute value, but we use two little lines on each side instead of one line.

When we draw a vector as an arrow in the xy-plane, the length of the arrow is the magnitude of the vector.

This length also happens to be the hypotenuse of a right triangle whose legs have lengths x and y. We can find the magnitude of the vector using the Pythagorean Theorem.

The magnitude is  .

Be Careful: When finding the magnitude of a vector with negative components, either 

  • use parentheses and square the components correctly, or
  • drop the negative signs altogether.

Since finding magnitude requires squaring the components anyway, it doesn't matter if the components are negative or positive.

Magnitude is a lot like absolute value in this way.

Whether a number is positive or negative doesn't affect its absolute value: |-5| = |5|.

Similarly, whether the components of a vector are positive or negative doesn't affect its magnitude: ||<1,2>|| = ||<-1,2>|| = ||<1,-2>|| = ||<-1,-2>||.

Look at the vectors on a graph. The arrows are all the same length; they're pointing different directions:

If a vector lies flat along the x- or y-axis, we don't need to use the Pythagorean Theorem to find its magnitude. Such a vector will have only one component that isn't zero, and that component will be the magnitude of the vector.