Vector Operations Lesson

Vectors Operations - Notes

• Click through the slides and take notes as needed.

• Attempt the practice problems on your notes sheet.

• Answer the practice problems here to show mastery.

Vector Mastery

what you need to be able to do to master this standard

• The tail is the starting position.

• The head is the ending position.

• The magnitude represents how long or large the vector is.

  • For example, a vector representing 100 should be twice as long as a vector representing 50. ​

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​How To Draw a Vector

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Vector Operation Rules

Scalar Multiplication:

• "distribute" the scalar to all parts of the matrix

• Negative scalar will change the direction of the vector

• kv = <ka, kb>

Addition:

• add the corresponding parts

• v+w = (a1+a2, b1+b2)

Subtraction:

• subtract the corresponding parts

• v-w = (a1-a2, b1-b2)

Multiple Choice

Given c = <-4, 5> find 2c

• <-12, 4>

• <-2, 3>

• <-8, 10>

• <-2, 2.5>

Multiple Choice

If a = <-6, 2> and b = <1, 7>, find a + b.

• <-7, -5>

• <7, 9>

• <-5, 9>

• <-4, 8>

Multiple Choice

If a = <-6, 2> and b = <1, 7> and c = <-4, 5>, find c - b.

• <-5, -2>

• <5, 2>

• <-5, 2>

• <5, -2>

Multiple Choice

If a = <-6, 2> and b = <1, 7> and c = <-4, 5>, find a - 3c

• <-18, 17>

• <-2, -3>

• <6, -13>

• <5, -2>

Multiple Choice

If v =〈a,b〉, then a is the ___________ component of v.

• vertical

• horizontal

• parallel

• slope

Multiple Choice

If a = <-6, 2> and b = <1, 7> and c = <-4, 5>, find (1/2)a - b

• <-7/2, -5/2>

• <-4, -6>

• <4, 6>

• <-2, 8>

Multiple Choice

If a = <-6, 2> and b = <1, 7> and c = <-4, 5>, find 4b + 5a

• <-19, 43>

• <19, -43>

• <26, -38>

• <-26, 38>

A unit vector has a magnitude of 1.

In many applications of vectors, it is useful to find a unit vector that has the same direction as a given nonzero vector v.

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Unit Vectors

To find a unit vector, divide vector v by its length (magnitude).

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Unit Vectors

• Note that u is a scalar multiple of v

• Vector u has a magnitude of 1

• Vector u has the same direction as v

• Vector u is called a unit vector in the direction of v​

Unit Vectors

Find a unit vector in the direction of v = <-5,12>. Verify that the result has a magnitude of 1.

Multiple Choice

Find the unit vector in the direction of v = <5,12>.

• <5,12>

• $$<\frac{5}{13},\frac{12}{13}>$$  

• $$<\frac{5}{\sqrt[]{13}},\frac{12}{\sqrt[]{13}}>$$  

• $$<\frac{5}{\sqrt[]{2}},\frac{12}{\sqrt[]{2}}>$$  

Match

Match the vector to it's corresponding unit vector

• <-5, 12>

• <9, 12>

• <-4, -1>

• <16, -8>

• $$<-\frac{5}{13},\ \frac{12}{13}>$$

• $$<\frac{3}{5},\ \frac{4}{5}>$$

• $$<-\frac{4\sqrt[]{17}}{17},-\frac{\sqrt[]{17}}{17}>$$

• $$<\frac{2\sqrt[]{5}}{5},\ -\frac{\sqrt[]{5}}{5}>$$

Multiple Choice

Find the unit vector n the direction of v = <6,6>.

• $$<1,\ 1>$$

• $$<\frac{\sqrt[]{3}}{2},\frac{1}{2}>$$  

• $$<\frac{1}{2},\ \frac{\sqrt[]{3}}{2}>$$

• $$<\frac{\sqrt[]{2}}{2},\ \frac{\sqrt[]{2}}{2}>$$  

Multiple Choice

Find the unit vector n the direction of v = <9,-6>.

• <3,-2>

• $$<\frac{3\sqrt[]{13}}{13},\ \frac{-2\sqrt[]{13}}{13}>$$  

• $$<\frac{3}{13},-\frac{2}{13}>$$  

• $$<\frac{-2}{\sqrt[]{13}},\frac{3}{\sqrt[]{13}}>$$  

Multiple Select

What are the standard unit vectors. Select all that apply.

• <0, 1>

• <1, 1>

• <0,0>

• <1,0>

Multiple Choice

What is the horizontal component of $$v=<v_1,v_2>$$  

• $$v_1$$  

• $$v_2$$  

• none of these

Multiple Choice

What is the vertical component of $$v=<v_1,v_2>$$  

• $$v_1$$  

• $$v_2$$  

• none of these

Multiple Choice

Write $$v=<a,\ b>$$   as a linear combination of i and j.

• $$a\ +\ b$$  

• $$ai\ +\ bi$$  

• $$ai\ +bj$$  

• $$aj\ +bi$$  

Multiple Choice

Write vector v = <2, 5> as a linear combination of standard unit vectors.

• <2, 5>

• 2 + 5

• 2i + 5j

• 2a + 5b

Multiple Choice

Write vector v = <-7, 3> as a linear combination of standard unit vectors.

• -7i + 3j

• 3i - 7j

• <-7i, 3j>

• -7a + 3b

Let RS be the vector with R(-9,8) and S(-5,-6). Write RS as a linear combination of the standard unit vectors.

Fill in the Blank

Let RS be the vector with R(-9,8) and S(-5,-6). Write RS as a linear combination of the standard unit vectors.

Multiple Choice

Let PQ be the vector with P (-10,6) and Q (-2,5). Write PQ as a linear combination of the standard unit vectors..

• -12i + 11j

• 8i - j

• -8i + j

• 12i-11j

​This is the end of the notes for today. We will continue the rest of the notes in class on Thursday.

Complete the classwork assignment for these notes #1-12.

End of Notes