I need more examples and solution under dot multiplication and cross multiplication in application of vector.

Here are examples for dot product and cross product applications: Dot Product Application: Finding the Angle Between Two Vectors Problem: Find the angle between the vectors u = 3i + 2j - k and v = i - 4j + 2k. Step 1: Calculate the dot product u · v. The dot product of two vectors u = u_xi + u_yj + u_zk and v = v_xi + v_yj + v_zk is given by u · v = u_xv_x + u_yv_y + u_zv_z. u · v = (3)(1) + (2)(-4) + (-1)(2) u · v = 3 - 8 - 2 u · v = -7 Step 2: Calculate the magnitudes of u and v. The magnitude of a vector u = u_xi + u_yj + u_zk is |u| = sqrt(u_x^2 + u_y^2 + u_z^2). |u| = sqrt(3^2 + 2^2 + (-1)^2) = sqrt(9 + 4 + 1) = sqrt(14) |v| = sqrt(1^2 + (-4)^2 + 2^2) = sqrt(1 + 16 + 4) = sqrt(21) Step 3: Use the dot product formula to find the angle. The formula relating the dot product, magnitudes, and angle is u · v = |u| |v| . Rearranging for : = u · v|u| |v| = (-7)/(sqrt(14) 21) = (-7)/(sqrt(294)) = (-7)/(7sqrt(6)) = (-1)/(sqrt(6)) Now, find : = ((-1)/(sqrt(6))) ≈ (-0.4082) ≈ 114.09^ Cross Product Application: Finding a Perpendicular Vector and Area of a Parallelogram Problem: Given vectors a = i + 2j - k and b = 3i - j + 2k, find a vector perpendicular to both a and b, and then find the area of the parallelogram formed by these two vectors. Step 1: Calculate the cross product a × b. The cross product of two vectors a = a_xi + a_yj + a_zk and b = b_xi + b_yj + b_zk is given by: a × b = i & j & k \\ a_x & a_y & a_z \\ b_x & b_y & b_z a × b = i & j & k \\ 1 & 2 & -1 \\ 3 & -1 & 2 a × b = i((2)(2) - (-1)(-1)) - j((1)(2) - (-1)(3)) + k((1)(-1) - (2)(3)) a × b = i(4 - 1) - j(2 - (-3)) + k(-1 - 6) a × b = 3i - 5j - 7k A vector perpendicular to both a and b is 3i - 5j - 7k. Step 2: Calculate the magnitude of the cross product to find the area. The area of the parallelogram formed by a and b is equal to the magnitude of their cross product, |a × b|. |a × b| = |3i - 5j - 7k| |a × b| = sqrt(3^2 + (-5)^2 + (-7)^2) |a × b| = sqrt(9 + 25 + 49) |a × b| = sqrt(83) The area of the parallelogram is sqrt(83) square units. Send me the next one 📸