Question

Vectors u and v are shown on the graph.Part A: Write u and v in component form. Show your work. (3 points)Part B: Find u + v. Show your work. (2 points)Part C: Find 5u − 2v. Show your work. (5 points)

Answer 1

`\begin{gathered} (A)u<9,7>;v<8,-2> \\ (B)<17,5> \\ (C)<29,39> \end{gathered}`

Step-by-step explanation

(A) We want to write the vectors in component form.

To do this, we will write them in terms of their horizontal and vertical lengths.

To do this, we have to find the difference between the coordinate points of the endpoints and the starting points of the vectors.

Hence, for vector u, its starting point is (2, -6) and its endpoint is (11, 1).

Hence, its component form is:

`\begin{gathered} <11,1>-<2,-6> \\ \Rightarrow<11-2,1-(-6)> \\ \Rightarrow<9,1+6> \\ <9,7> \end{gathered}`

For the vector v, its starting point is (5, 8) and its endpoint is (13, 6).

Hence, its component form is:

`\begin{gathered} <13,6>-<5,8> \\ \Rightarrow<13-5,6-8> \\ \Rightarrow<8,-2> \end{gathered}`

Hence, the component form of the vectors is:

`\begin{gathered} u<9,7> \\ v<8,-2> \end{gathered}`

(B) To find the sum of the two vectors, we have to find the sum of their components.

That is:

`\begin{gathered} u+v=<9,7>+<8,-2> \\ \Rightarrow<9+8,7+(-2)> \\ <17,7-2> \\ <17,5> \end{gathered}`

That is the answer.

(C) To find 5u - 2v, we have to multiply vector u by 5, multiply vector v by 2, and then find the difference:

`\begin{gathered} 5\lbrack<9,7>\rbrack-2\lbrack<8,-2>\rbrack \\ \Rightarrow<45,35>-<16,-4> \\ \Rightarrow<45-16,35-(-4)> \\ \Rightarrow<29,35+4> \\ \Rightarrow<29,39> \end{gathered}`

That is the answer.