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.

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