need help here please6.041. Find a ⋅ b.a = <7, 4>, b = <3, 5> (2 points) 41 <10, 9> -1 <21, 20>2. Find a ⋅ b.a = 10i + 9j, b = 4i + 3j (2 points) <40, 27> <14, 12> 67 -133. Find the angle between the given vectors to the nearest tenth of a degree.u = <6, -1>, v = <7, -4> (2 points) 20.3° 10.2° 0.2° 30.3°4. Determine whether the vectors u and v are parallel, orthogonal, or neither.u = <10, 6>, v = <9, 5> (2 points) Orthogonal Parallel Neither5. Evaluate the expression:v ⋅ wGiven the vectors:r = <8, 8, -6>; v = <3, -8, -3>; w = <-4, -2, -6> (2 points)

1. Find a ⋅ b.
a = <7, 4>, b = <3, 5> (2 points)

41 <---------- right answer
<10, 9>
-1
<21, 20>

Explanation:

a . b means the dot or scalar product of the vectors a and b.

The scalar product of vectorr <x1,y2> and <x2,y2> is (x1)*(y1) + (x2)*(y2)

So, a . b = 7*3 + 4*5 = 21 + 20 = 41 <-------- answer


2. Find a ⋅ b
a = 10i + 9j, b = 4i + 3j (2 points)

<40, 27>
<14, 12>
67              <---------- answer
-13

Explanation:
Again you are required to find the dot product of two vectors. In this case the vectors are given using the representation with the unit vectors i and j.

The dot product of the vectors (x1 i + y1 j) . (x2 i + y2 j) is x1*x2 + y1*y2

So, (10i + 9j) . (4i + 3j) = 10*4 + 9*3 = 40 + 27 = 67

Answer: 67


3. Find the angle between the given vectors to the nearest tenth of a degree.
u = <6, -1>, v = <7, -4> (2 points)

20.3°   <--------------- answer
10.2°
0.2°
30.3°

Explanation:

You can use the dot product to find the angle between two vectors.

This is the formula:

cos(α) = [ <a> dot <b. ] / [|a| |b| ]

where α is the angle between the vector a and b.

<a> is the vector a, <b> is the vector b, <a> dot <b> is the dot product, |a| is the magnitude of vector a, and |b| is the magnitude of vector b.

<u> dot <v> = (6)(7) + (-1)(-4) = 42 + 4 = 46

|u| = √ (6^2 + 1^2) = √37
|v| = √ (7^2 + 4^2) = √65

=> cos(α) = [46] / [√37 * √65] = 0.938

=> α = arccos(0.938) = 20.3°


4. Determine whether the vectors u and v are parallel, orthogonal, or neither.
u = <10, 6>, v = <9, 5> (2 points)

Orthogonal
Parallel
Neither  <------------ answer

Explanation:

At sight they are neither orthogonal nor parallel. You can put the points in a graph and you will realize inmediately.

Let's calculate the angle, with the formula given in the above problem.

cos(α) = [<u> dot< >v] / [ |u| |v| ]

<u> dot <v> = 10*9 + 6*5 = 90 + 30 = 120

|u| = √(10^2 + 6^2) = √136
|v| = √(9^2 + 5^2) = √106

cos(α) = 120 / (√136 * √106] = 0,999

α = arctan(0,999) = 1,9°

Which confirms that they are neither orthogonal nor parallel


5. Evaluate the expression:

v ⋅ w

Given the vectors:

r = <8, 8, -6>; v = <3, -8, -3>; w = <-4, -2, -6> (2 points)

Solution:

v . w = v dot w = (3)(-4) + (-8)(-2) + (-3)(-6) = -12 + 16 + 18 = 22

Answer: 22