How to calculate for vector a minus b and also a dot b and also a minus b

To explain how to calculate these, let's assume we have two general vectors: a = a_xi + a_yj + a_zk b = b_xi + b_yj + b_zk Here's how to perform each calculation: 1. Vector Subtraction: a - b To subtract vectors, you subtract their corresponding components. Step 1: Subtract the i components. Step 2: Subtract the j components. Step 3: Subtract the k components. a - b = (a_x - b_x)i + (a_y - b_y)j + (a_z - b_z)k For example, if a = 3i + 2j + k and b = i - 4j + 5k: a - b = (3 - 1)i + (2 - (-4))j + (1 - 5)k a - b = 2i + 6j - 4k The result is a new vector. 2. Dot Product: a · b The dot product (also called scalar product) results in a scalar (a single number), not a vector. You multiply corresponding components and then sum the results. Step 1: Multiply the i components. Step 2: Multiply the j components. Step 3: Multiply the k components. Step 4: Sum these products. a · b = (a_x)(b_x) + (a_y)(b_y) + (a_z)(b_z) For example, if a = 3i + 2j + k and b = i - 4j + 5k: a · b = (3)(1) + (2)(-4) + (1)(5) = 3 - 8 + 5 = 0 The result is a scalar: 0. 3. Cross Product: a × b The cross product (also called vector product) results in a new vector that is perpendicular to both a and b. It is calculated using a determinant. Step 1: Set up the determinant matrix. a × b = i & j & k \\ a_x & a_y & a_z \\ b_x & b_y & b_z Step 2: Expand the determinant. a × b = i(a_y b_z - a_z b_y) - j(a_x b_z - a_z b_x) + k(a_x b_y - a_y b_x) For example, if a = 3i + 2j + k and b = i - 4j + 5k: a × b = i & j & k \\ 3 & 2 & 1 \\ 1 & -4 & 5 = i((2)(5) - (1)(-4)) - j((3)(5) - (1)(1)) + k((3)(-4) - (2)(1)) = i(10 - (-4)) - j(15 - 1) + k(-12 - 2) = i(14) - j(14) + k(-14) = 14i - 14j - 14k The result is a new vector: 14i - 14j - 14k. Send me the next one 📸