8.5.3 * Unit Vectors, Linear Combinations

8.5.3 – Unit Vectors, Linear
Combinations
• In the case of vectors, we have a special vector
known as the unit vector
– Unit Vector = any vector with a length 1; direction
irrelevant
• Two special unit vectors we look at the most;
– i = {1, 0}
– j = {0, 1}
• What would vector i represent?
• What would vector j represent?
• Regardless of the vector, any vector may be
written in terms of the vector i and j
• {a, b} = a{1,0} + b{0, 1} = ai + bj
– Known as a Linear Combination
– LC = sum of scalar multiples of vectors
Finding LC
• To find linear combinations of vectors in terms
of a select unit vector, and vector i and j;
• The scalar a will be represented by;
– a = 1/|| u ||
– This will give us the unit vector in the same
direction as a given vector u
• To write the linear combination, just take out
the horizontal/vertical component of the
component form
• Example. If u = {-5, 3}, then u = -5i + 3j
• Example. Let u = {6. -3}. Find a unit vector
pointing in the same direction as u.
• Example. Write the given vector as a linear
combination of i and j.
• Example. Let u = {-4, -8}. Find a unit vector
pointing in the same direction as u.
• Example. Write the given vector as a linear
combination of i and j.
• Example. Let u = {2, 3}. Find a unit vector
pointing in the same direction as u.
• Example. Write the given vector as a linear
combination of i and j.
• Assignment
• Pg. 667
• 27-32