A vector v = [3, 8] has a rectangular component in the direction [3, 1]. Find the rectangular components of vector v. use grade 12 knowledge.
Question
A vector v = [3, 8] has a rectangular component in the direction [3, 1]. Find the rectangular components of vector v. Use grade 12 knowledge.
Solution
To find the rectangular components of a vector in a specific direction, we first need to find the unit vector in the direction of [3, 1].
Step 1: Find the magnitude of the direction vector [3, 1] The magnitude (or length) of a vector a = [a1, a2] is given by √(a1^2 + a2^2). So, the magnitude of the vector [3, 1] is √(3^2 + 1^2) = √10.
Step 2: Find the unit vector in the direction of [3, 1] The unit vector in the direction of a vector a = [a1, a2] is given by [a1/magnitude, a2/magnitude]. So, the unit vector in the direction of [3, 1] is [3/√10, 1/√10] = [0.9487, 0.3162].
Step 3: Find the component of v in the direction of [3, 1] The component of a vector v = [v1, v2] in the direction of a unit vector u = [u1, u2] is given by (v . u) * u, where "." denotes the dot product. The dot product of v and u is v1u1 + v2u2. So, the dot product of v = [3, 8] and u = [0.9487, 0.3162] is 30.9487 + 80.3162 = 4.3421. Therefore, the component of v in the direction of [3, 1] is 4.3421 * [0.9487, 0.3162] = [4.12, 1.37].
Step 4: Find the component of v orthogonal to [3, 1] The component of a vector v = [v1, v2] orthogonal to a unit vector u = [u1, u2] is given by v - (v . u) * u. We already found that (v . u) * u = [4.12, 1.37], so the orthogonal component is [3, 8] - [4.12, 1.37] = [-1.12, 6.63].
So, the rectangular components of the vector v = [3, 8] in the direction of [3, 1] are [4.12, 1.37] and [-1.12, 6.63].
Similar Questions
Given the following vector v:v=⟨8,1⟩𝑣=⟨8,1⟩Step 1 of 2 : Find a unit vector pointing in the same direction as v.
If v lies in the first quadrant and makes an angle 𝜋/3 with the positive x-axis and |v| = 10, find v in component form.
Find the component form of the vector originating from 3, −1 with terminal point at −5, 6
What is the resultant vector if a vector 7 units from the origin along the X axis is added to a vector 3 units along the Y axis?
Vector R has a magnitude of 36 units directed 30˚ from the +y – axis of a Cartesian plane. What are the x- and y-components of the vector?
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.