INSTRUCTOR: Let vector a equal, angle bracket, 7, comma, 1, angle bracket. And let the vector b be the vector with initial point 3, 2 and terminal point negative 1, negative 1. Part a, find the magnitude of a. The magnitude of a vector is the sum of the squares of its components, and then we take the square root of that result. So the magnitude, given by two vertical bars of vector a, is equal to the square root of 7 squared plus 1 squared.

Now, 7 squared is 49. 1 squared is 1. The sum of those two is 50, so this is the square root of 50. Now, because 50 is 25, a perfect square, times 2, the square root of 50 is the square root of 25 times 2, which can be simplified to be 5 square root of 2. So the magnitude of a is equal to 5 root 2.

Part b. We want to express vector b in component form. That is, with the angle bracket notation. So because we are given two points, the initial point and the terminal point, we can write this in component form by subtracting the terminal point, the components of that, doing the terminal point minus the initial point.

That's what we're going to do. We're going to subtract those two points, and that will give us the components of our vector. So vector b equals, angle bracket, negative 1, minus 3, comma, negative 1, minus 2, the values in those points. So that this will be, angle bracket, minus 4, comma, negative 3, angle bracket. And that is our vector b.

Part c. Find 3a minus 4b. 3a minus 4b will equal to 3. Angle bracket, 7, comma, 1, angle bracket. That's vector a. Minus 4, angle bracket. And we have negative 4, comma, negative 3, closing angle bracket for vector b. Now, multiplication by scalars with these means we can just multiply each component by the scalar.

So the first of those two products would be, angle bracket, 21, comma, 3, closing angle bracket, minus-- and multiplying a 4 in there. That would be a negative 16. So angle bracket, negative 16, comma, negative 12, closing angle bracket.

Now, for subtraction of two vectors, we can subtract these component-wise. So we'll have 21 minus negative 16, which will be 37. So angle bracket, 37, comma. And then we have 3 minus negative 12, which would be 15. And then closing angle brackets. So we have a result of 3a minus 4b equals the vector 37, 15.