Dot Product

Question 1

Dot Product Theorem

Answer 1

GIven two vectors a and b, the dot product of the two is the product of all the respetive terms, then the sum of all

Question 2

Dot product equations

Answer 2

• a * b = a1b1 + a2b2 + a3b3
• a * b = |a| |b| cos (theta)

Question 3

vectors a and b are perpendicular

Answer 3

when a * b = 0 and/or theta is 90 degrees

Question 4

cos is positive (I or IV Quadrant)

Answer 4

when theta is acute (<90)

Question 5

cos is negative (II or III Quadrant)

Answer 5

when theta is obtuse (>90)

Question 6

vectors a and b are parallel

Answer 6

when b1/a1 = b2/a2 = b3/a3

Question 7

Dot Product Properties

Answer 7

1) a * a = (a1a1 + a2a2 + a3a3) or (|a| |a| cos (0) )= |a| |a| (dot product of itslef is length squared; angle between a & a is cos (0) which is 1;

2) 0 * a = 0 ( dot product of a vector and vector “0” is the scalar number 0)

3) 0a = 0 (product of scalar number and vector is another vector)

4) (|b| cos (theta)) ^2 + (|b| sin (theta)) ^2 = |b| |b|

Question 8

Scalar or Component Projection of b over a

Answer 8

(results in scalar number) |b|cos (theta) = (a * b)/ |a|

Question 9

Vector Projection or Projection of b over a

Answer 9

(reuslts in vecor) (a * b)/ |a| times (a/|a|) = (a * b)/|a|^2 times vector a

Question 10

when b is smallar than a (in terms of projections)

Answer 10

scalar project would be less than one (fraction)

Question 11

when b is bigger than a (in terms of projections)

Answer 11

scalar projection will be more than one