This English question involves literary analysis, grammar, or writing skills. The detailed response below provides a well-structured answer with supporting evidence and clear explanations.
ScanSolve AI Answer
Vectors are quantities that have both magnitude (size or length) and direction. They are often represented by arrows. The magnitude* of a vector a is its length, written as ||a||. A unit vector* is a vector with a magnitude of 1. Vectors can be added, subtracted, and multiplied by a scalar (a regular number). In 3D, a vector a can be written in component form as a = a_xi + a_yj + a_zk. Its magnitude is ||a|| = sqrt(a_x^2 + a_y^2 + a_z^2). The dot product* a · b results in a scalar (a number). It's used to find the angle between two vectors or to check if they are perpendicular (if a · b = 0). The cross product* a × b results in a new vector that is perpendicular to both a and b. Its magnitude equals the area of the parallelogram formed by a and b. It's used to check if vectors are parallel (if a × b = 0). The scalar triple product a · (b × c) gives the volume of the parallelepiped formed by the three vectors. If this product is zero, the vectors are coplanar* (they lie on the same plane). --- Example 1: Given a = 2i - 3j + k and b = i + 2j - 3k. a) Find a + b Step 1: Add the corresponding components of a and b. a + b = (2i - 3j + k) + (i + 2j - 3k) a + b = (2+1)i + (-3+2)j + (1-3)k a + b = 3i - j - 2k The sum of the vectors is 3i - j - 2k. b) Find a - b Step 1: Subtract the corresponding components of b from a. a - b = (2i - 3j + k) - (i + 2j - 3k) a - b = (2-1)i + (-3-2)j + (1-(-3))k a - b = i - 5j + 4