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
The scalar dot product of two vectors a and b is a scalar quantity. It is calculated as the product of their magnitudes and the cosine of the angle between them. Formula: a · b = |a| |b| In terms of components, if a = a_1, a_2, a_3 and b = b_1, b_2, b_3 , then: a · b = a_1 b_1 + a_2 b_2 + a_3 b_3 Example: Let a = 1, 2, 3 and b = 4, 5, 6 . a · b = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32 The vector cross product of two vectors a and b is a vector quantity that is perpendicular to both a and b. Its magnitude is the product of their magnitudes and the sine of the angle between them. Formula: a × b = |a| |b| n where n is a unit vector perpendicular to the plane containing a and b, determined by the right-hand rule. In terms of components, if a = a_1, a_2, a_3 and b = b_1, b_2, b_3 , then: a × b = i & j & k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 = (a_2 b_3 - a_3 b_2) i - (a_1 b_3 - a_3 b_1) j + (a_1 b_2 - a_2 b_1) k Example: Let a = 1, 2, 3 and b = 4, 5, 6 . a × b = i & j & k \\ 1 & 2 & 3 \\ 4 & 5 & 6 = (2 · 6 - 3 · 5) i - (1 · 6 - 3 · 4) j + (1 · 5 - 2 · 4) k = (12 - 15) i - (6 - 12) j + (5 - 8) k = -3 i + 6 j - 3 k = -3, 6, -3 Send me the next one 📸