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.

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].