>> Now we'll look at two examples of determining the magnitude and direction
of a vector given in component form.
This first example we're going to sketch the vector and then use what we know
about right triangles to determine the magnitude and direction.
For the second example we use formulas for the magnitude
and direction that can save us some time.
So, for the first example, our vector has an initial point at the origin
and a terminal point at the point 3, 4.
The terminal point would be here, so here's our vector.
The direction would be this angle here between the positive x axis and the vector.
The horizontal component is 3, and the vertical component is 4.
So, the magnitude of vector u would be the length of this vector,
and because we have a right triangle, we can use the Pythagorean Theorem
to determine the magnitude of u. -- - We would have the magnitude of u squared,
is equal to 3 squared, plus 4 squared [writing].
So, we have the magnitude of u squared, is equal to 3 squared is 9,
4 squared is equal to 16, 9 plus 16 is 25.
So, now we would take the square root of both sides of the equation,
magnitude is always positive, so we have the magnitude of u,
equals the square root of 25, or 5.
And now, going back to our right triangle, we can determine our direction,
or angle theta by using the tangent function which is the ratio of the opposite side,
to the adjacent side -- - so we know tangent theta must equal 4/3.
Now, having a tangent function value of 4/3,
the final value that we'll find using the unit circle of our reference triangles
which means we'll have to use the calculator
to get a decimal approximation for our direction theta.
So, if tangent theta equals 4/3 that means theta is equal to arch tangent or inverse tangent 4/3,
and now we'll go to the calculator to get an approximation for theta.
We do want to give our angle and degrees, so let's make sure
that we're in degree mode, we are.
So, we'll go back to the home screen, press second tangent 4/3, close parenthesis.
So, the direction of the vector is approximately 53.1 degrees.
-- - So, here's the magnitude of the vector, and here's the direction of the vector.
Now for our second example, we'll use the formulas seen in most textbooks.
If we're give a vector u, in component form, the magnitude of u is equal
to the square root of x squared, plus y squared.
This is just the Pythagorean Theorem, and tangent theta is equal to y divided by x. Again,
we used both of these formulas in the first example,
we just developed them ourselves using the right triangle.
So, for the second example the vector has a terminal point of negative 6,
3 which would be here [writing], and again, we want to find the magnitude
of vector u [writing], and the direction which would be this angle here.
But, this time, let's use our formulas.
So, we have the magnitude of u, equals the square root of x squared, plus y squared,
or negative 6 squared, plus 3 squared, put the magnitude of u,
is equal to negative 6 squared is 36, plus 3 squared is 9, 36 plus 9 is equal
to 45, for the square root of 45.
But, this does simplify, it has a perfect square factor of 9, and since the square root
of 9 is equal to 3, we have a magnitude of u, is equal to 3 square root, 5.
And now, to find the direction, we know that tangent theta must equal y divided by x
which would be 3 divided by negative 6, or negative 1/2.
So if tangent theta equals negative 1/2 then theta equals inverse tangent of negative 1/2.
Now, we'll go to the calculator but,
keep in mind that our angle theta must be in the second quadrant.
So, we have second tangent negative 1/2.
[ Silence ]
Notice how the calculator's giving us approximately negative 26.6 degrees.
And this angle is in the fourth quadrant but, that's okay because, we can still use this angle
to determine our angle which we know is in the second quadrant.
-- So, the calculator gave us this angle here which is approximately negative 26.6 degrees,
the tangent is also negative in the second quadrant
which means our angle must have a reference angle of 26.6 degrees in the second quadrant.
So, if this is 26.6 degrees, then we can find our angle,
and that would be 180 degrees minus 26.6 degrees.
So, our direction, or angle theta is approximately 153.4 degrees [writing].
So, this example is a good reminder that even if we do use the shortcut formulas,
it is important to sketch the vector to make sure
that our angle theta is in the correct quadrant.
Here's our magnitude, and here's our direction.
I hope you found these two examples helpful. ------------------------------8109e34d137c--