Vector

Most of the vectors are used to find the coordinates of apoint position.Not only that,we can also use vectors to find the speed at which something is moving or increasing.

​

​Group 2

​What is Vector

​Change The Point To The Vector

​Question:-

a. A(3,3), B(-2, 2) b. A (8, 0, -2), B(5, 3, 1)

c. A(-1,8), B(5, 9) d. A (0, 7, 5), B(11, 5, 0)

Multiple Choice

Find the $$\overrightarrow{XY}$$

If the X(12, -13 , -4) and Y(20 , -6 , 0)

• (6 , 7, 4)

• (8 , 7, 4)

• (8 , -19, -4)

• (-6 , -7, -4)

Multiple Choice

Change the point to be vector , $$\overrightarrow{GP}$$ .

If the points is G(3, 0) and P(2, 7)

• (-1 , 7)

• (1 , -7)

• ( 1 , -7)

​Magnitude or length

​The length of 2-dimensional vector a =

The length of 3-dimensional vector a =

​Find the Magnitude

1.A(1 , 2 , 3) and B(3 , 3 , 4)

​2.X(1 , 3 ) and Y(1 , 4 )

​is

is

Multiple Choice

Find the $$\left|a\right|$$ ,When A⟨2,3⟩ and B ⟨-2,1⟩

• -4.4721

• 4.4721

• Cannot be calculated

• 4.4722

Multiple Choice

Find the magnitude for the vector a=⟨6,0,3⟩

• $$\sqrt[]{5}$$

• $$\sqrt[]{1}$$

• 3.2361

• -3.2361

​Vector Addition

​If a=

​and b=

​,then

​and

​A)Given that a=6i -j +3k ,b=5j + k

1.a+b

​2.a-b

​1. (6i - j + 3k) + (0i + 5j + k)

=(6+0)i +(-1+5)j +(3+1)k

=6i + 4j + 4k

​2.(6i - j + 3k) - (0i + 5j + k)

=(6-0)i +(-1-5)j +(3-1)k

=6i + (-6)j + 2k

=6i - 6j + 2k

​B)Given that a=⟨-4 ,3) ,(6, 2)

1. 3a

​2.-4a + 5b

​1.3a=(-4(3) , 3(3) )

=(-12 , 9)

2.-4a + 5b =(-4(-4) ,3(-4)) + (6(5) , 2(5))

=(16 , -12) + (30 , 10)

=(46, -2)

Multiple Choice

If a=⟨6,0,3⟩ ,b=⟨-1,5,-2⟩.

Find the $$\frac{1}{2}a+2b$$

• (1 , 10 , $$-\frac{5}{2}$$ )

• (10, 10 , $$\frac{5}{2}$$ )

• (1, 10 , $$\frac{5}{2}$$ )

• (10 ,10 , $$-\frac{5}{2}$$ )

Multiple Choice

If a=i + 6j - k, b=7i - k

Find the -4a + 5b

• 31i -24j -k

• 31i + 24j -k

• -31i -24j +k

• 31i - 24j +k

Multiple Choice

If a=2i - j + 2k , b=4j+2k

Find the $$\left|a\right|$$ and $$\left|b\right|$$

• $$\left|a\right|$$ = 3

  $$\left|b\right|$$ = $$\sqrt[]{20}$$

• $$\left|a\right|$$ = $$\sqrt[]{29}$$

  $$\left|b\right|$$ = $$\sqrt[]{20}$$

• $$\left|a\right|$$ =3

  $$\left|b\right|$$ = $$\sqrt[]{29}$$

• $$\left|a\right|$$ =3

  $$\left|b\right|$$ =2

​Algebraic Properties of Vectors

• Commutative (vector) P + Q = Q + P

• Associative (vector) (P + Q) + R = P + (Q + R)

• Additive identity There is a vector 0 such that (P + 0) = P = (0 + P) for all P

• Additive inverse For any P there is a vector -P such that P + (-P) = 0

• Distributive (vector) r(P + Q) = rP + rQ

• Distributive (scalar) (r + s) P = rP + sP

• Associative (scalar) r(sP) = (rs)P

• Multiplicative identity For the real number 1, 1P = P for each P

​Additional For Basic Knowlegde

​Unit vector

• A vector of length 1 is called a unit vector.

• In an xy -coordinate system the unit vector is denoted by i and j

• In an xyz -coordinate system the unit vector is denoted by i, j and k

​Dot Product

​If a=

​and b=

​then their dot product is:

​This also applies to 3-space vectors

​Find the Dot product

1.a=⟨4, -1⟩ ,b=⟨3, 6⟩

​2. a=i -2j +3k ,b=5i + 9k

=4(3) + 6(-1)

=6

​Properties of the Dot Product

• (Commutative Property) For any two vectors A and B, A^.B = B^.A.

• (Scalar Multiplication Property) For any two vectors A and B and any real number c, (cA)^.B = A^.(cB) = c(A^.B)

• (Distributive Property) For any 3 vectors A, B and C, A^.(B+C) = A^.B + A^.C.

​Additional information

=5(1) + 0(-2) + 9(3)

=5+0+27

=32

​1.

​2.

Multiple Choice

Find the Dot Product.

If a=⟨5 ,0, 2⟩ , b=⟨3, -1 , 10⟩

• -5

• 5

• 35

• -35

Multiple Choice

Find the Dot Product.

If a=2i + 9j ,b=4i - 3k

• 8

• -8

• 4

• -4

​Angle Between The Vectors

​The angle between two vectors a and b is found using the formula

​Find the angle between the vectors

1.a=3i + 4j – k and b=2i – j + k.

​First Step :Find the Dot Product

= (3i + 4j – k).(2i – j + k)

= (3)(2) + (4)(-1) + (-1)(1)

= 6-4-1

= 1

​Last Step:put all the values you just found into the Formula

Multiple Choice

Find the between angle the vectors.

If a=(1 , 2 ,3) and b(4 ,0 ,-1).Giving your answer in 4 decimal places and radian

• 1.5059rad

• -1.5059rad

• 1.6067rad

• -1.6067rad

Multiple Choice

Find the angle between the vectors.

If a=j + k , b= i + 2j - 3k. Giving your answer in 4 decimal places and radian

• 1.7609rad

• -1.7609rad

• 100.89rad

• -100.89rad

​Cross Product

​this are for three-dimensional vectors only.

if the a=

​and b=

​then the cross product is

​Question of Cross Product

​1.Find the cross product for a=⟨1,2,3⟩ b=⟨1,2,3⟩

​2.Find the cross product for a=2i + j - k ,b=j + 2k

​i j k

​2 1 -1

​0 1 2

​=(2-(-1))i - (4-0)j + (2-0)k

=3i -4j + 2k

=⟨3 ,-4 , 2⟩

Multiple Choice

Find the Cross product $$A\times B$$

if a=2i + 6j -4k , b=-3i -9j + 6k

• ⟨0 , 0 ,0⟩

• ⟨72 ,-24 ,0⟩

• -24j+36k

• 72i -24j - 36k

Multiple Choice

Find the Cross product $$A\times B$$

If a=⟨1 , 2 ,0⟩ , b=⟨0 , 3 ,1⟩

• ⟨1 , 1 ,3⟩

• 2i -j +3k

• ⟨0 , -1 ,3⟩

• i + j- 3k

Multiple Choice

Find the $$A\times B$$

If $$\left|a\right|$$ =12 , $$\left|b\right|$$ =15 and the angle between a and b is $$\frac{\pi}{6}$$

• $$155.88\degree$$

• $$155.89\degree$$

• $$168.34\degree$$

• $$122.78\degree$$