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 more examples to illustrate the different types of vectors. For some types, the concept is clear with one or two examples, and providing five distinct ones would be repetitive. 1. Zero Vector: A zero vector has all its components equal to zero. 0 = 0 \\ 0 0 = 0 \\ 0 \\ 0 2. Unit Vector: A vector with a magnitude of 1. For a = 3 \\ 4 , its magnitude is ||a|| = sqrt(3^2 + 4^2) = 5. The unit vector is a = (1)/(5) 3 \\ 4 = (3)/(5) \\ (4)/(5) . For b = 0 \\ -7 , its magnitude is ||b|| = sqrt(0^2 + (-7)^2) = 7. The unit vector is b = (1)/(7) 0 \\ -7 = 0 \\ -1 . For c = 1 \\ 2 \\ 2 , its magnitude is ||c|| = sqrt(1^2 + 2^2 + 2^2) = sqrt(1+4+4) = sqrt(9) = 3. The unit vector is c = (1)/(3) 1 \\ 2 \\ 2 = (1)/(3) \\ (2)/(3) \\ (2)/(3) . 3. Position Vector: Represents the position of a point from the origin. Position vector of point P(2, -1, 3) is OP = 2 \\ -1 \\ 3 . Position vector of point Q(-5, 0) is OQ = -5 \\ 0 . 4. Displacement Vector: Represents the change in position from one point to another. From A(1, 2, 3) to B(4, 0, -1): AB = 4-1 \\ 0-2 \\ -1-3 = 3 \\ -2 \\ -4 From C(0, 0) to D(6, -8): CD = 6-0 \\ -8-0 = 6 \\ -8 5. Equal Vectors: Two vectors are equal if they have the same magnitude and direction (i.e., identical components). a = 2 \\ 5 and b = 2 \\ 5 Here, a = b. p = -1 \\ 0 \\ 7 and q = -1 \\ 0 \\ 7 Here, p = q. 6. Negative Vector: Has the same magnitude but opposite direction. If a = 3 \\ -2 \\ 1 , then -a = -3 \\ 2 \\ -1 . If d = -4 \\ 0 , then -d = 4 \\ 0 . 7. Collinear Vectors: Lie on the same line or are parallel. One is a scalar multiple of the other. Let a = 1 \\ 2 . If k=3, then b = 3a = 3 \\ 6 . a and b are collinear. Let e = 2 \\ -1 . If k=-2, then f = -2e = -4 \\ 2 . e and f are collinear. 8. Coplanar Vectors: Lie on the same plane. Three vectors a, b, and c are coplanar if one can be expressed as a linear combination of the other two. 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, so they are coplanar. Let u = 1 \\ 1 \\ 1 , v = 0 \\ 1 \\ 2 , and w = 1 \\ 3 \\ 5 . Here, w = 1u + 2v (since 1(1)+2(0)=1, 1(1)+2(1)=3, 1(1)+2(2)=5). Thus, u, v, and w are coplanar. What's next? 📸