You are watching: Find the unit vector having the same direction as v

Introduction: In this lesson, unit vectors and also their simple components will be defined and quantified. We will research both 2- and also 3-dimensional vectors. The Lesson: A unit vector is a vector which has a size of 1. because that example, the vector v = (1, 3) is not a unit vector due to the fact that
. The notation
represents the norm, or magnitude, of vector v. yet the vector w =
is a unit vector since
. The simple unit vectors room i = (1, 0) and also j = (0, 1) which space of length 1 and have directions follow me the positive x-axis and y-axis respectively. To discover a unit vector through the same direction together a provided vector, we divide by the size of the vector. For example, take into consideration the vector v = (1, 3) which has a magnitude of
. If we divide each component of v by
us will gain the unit vector uv which is in the exact same direction as v:
. Let"s Practice: We very first calculate
. We division each ingredient of r through
and we have:
We use scalar multiplication because 3i = (3, 0) and also – 2j = (0, – 2) and also we have actually v = 3i – 2j. In 3 dimensions, we have i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1) wherein each of these vectors is a unit vector pointing in the direction that the hopeful x-, y-, and z-axes respectively. as in instance (ii), we have actually v = i – 3j +4k. To discover a unit vector with the very same direction together v we very first calculate
.

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As such we have actually

If vector v = (3, – 7) what is a unit vector v the exact same direction as v?What is her answer?
 Vector w = (2, 4, – 1). Rewrite w together a mix of the straightforward vectors i, j, and k. What is a unit vector with the very same direction as w? What is your answer?