This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are mathematical examples for each type of vector: 1. Zero Vector: A zero vector in two dimensions is represented as: 0 = 0 \\ 0 In three dimensions: 0 = 0 \\ 0 \\ 0 2. Unit Vector: If we have a vector a = 3 \\ 4 , its magnitude is ||a|| = sqrt(3^2 + 4^2) = sqrt(9+16) = sqrt(25) = 5. The unit vector in the direction of a is: a = a||a|| = (1)/(5) 3 \\ 4 = (3)/(5) \\ (4)/(5) 3. Position Vector: The position vector of a point P(x, y, z) from the origin O(0, 0, 0) is: OP = x \\ y \\ z For example, the position vector of point P(2, -1, 3) is: OP = 2 \\ -1 \\ 3 4. Displacement Vector: If point A is (x_1, y_1, z_1) and point B is (x_2, y_2, z_2), the displacement vector from A to B is AB = OB - OA. For example, if A(1, 2, 3) and B(4, 0, -1): AB = 4-1 \\ 0-2 \\ -1-3 = 3 \\ -2 \\ -4 5. Equal Vectors: Two vectors a and b are equal if their corresponding components are equal. a = 2 \\ 5 and b = 2 \\ 5 Here, a = b. 6. Negative Vector: If a = 3 \\ -2 \\ 1 , its negative vector is: -a = -3 \\ -(-2) \\ -1 = -3 \\ 2 \\ -1 7. Collinear Vectors: Two vectors a and b are collinear if b = ka for some scalar k. Let a = 1 \\ 2 . If k=3, then: b = 3a = 3 1 \\ 2 = 3 \\ 6 Vectors a and b are collinear. 8. Coplanar Vectors: Three vectors a, b, and c are coplanar if one can be expressed as a linear combination of the other two (i.e., c = xa + yb for scalars x, y). Let a = 1 \\ 0 \\ 0 , b = 0 \\ 1 \\ 0 , and c = 2 \\ 3 \\ 0 . Here, c = 2a + 3b. All three vectors lie in the xy-plane (where z=0), so they are coplanar. 3 done, 2 left today. You're making progress.