8.6.1 – The Dot Product (Inner Product)

• So far, we have covered basic operations of vectors– Addition/Subtraction– Multiplication of scalars– Writing vectors in various forms

• We will now talk about the last crucial operation

Dot Product

• The product of two vectors will create a scalar• The dot product of two vectors is given if u =

{u1, u2} and v = {v1, v2}

•

• The dot product may be positive, negative, or zero (similar to multiplication of real numbers)

2211 vuvuvu

• Example. Find the dot product if u = {-5, 2} and v = {3, -1}

• Find each corresponding part

• Example. Find the dot product if u = {-5, 2} and v = {-5, 2}

• Example. Find the dot product if u = {-5,2} and v = {2, 5}

Properties

• With the dot product, we can derive certain properties

• 1) u . v = v . u (commutative) • 2) 0 . u = 0• 3) u . (v + w) = u . v + u . w (distribution)• 4) a(u . v) = (au) . v = u . (av) • 5) u . u = ||u||2

• Example. Find the quantity 3v . u if u = {-2, 3} and v = {4, 4}

• Example. Find the magnitude of the vector v if the dot product with itself is 12.

• Example. u . u = 80. Find ||u||.

Dot Product Theorem

• Similar to component form, we can talk about the dot product of vectors in terms of an angle

• Let u and v be nonzero vectors, and ϴ be the smaller of the two angles formed by u and v; then,

cos|||||||| vuvu

• Example. Find the angle between the two vectors u = {5,4} and v = {3, 2}

• Example. Find the angle between the two vectors u = 5i + 2j, v = 4i + j

• Assignment• Pg. 678• 1-23 odd