Let F1=10i-15j-20k,F2=6i+8j-12k. Find their dot product and the angle between those two vectors
student
25.33 deg
185,65. 66
To find the dot product of two vectors, F1 and F2, you need to multiply their corresponding components and add the results. The dot product is given by the formula:
F1 ⋅ F2 = (F1x * F2x) + (F1y * F2y) + (F1z * F2z)
Here, F1x, F1y, and F1z represent the x, y, and z components of vector F1, respectively. Similarly, F2x, F2y, and F2z represent the x, y, and z components of vector F2, respectively.
Let's calculate the dot product of the given vectors:
F1 ⋅ F2 = (10 * 6) + (-15 * 8) + (-20 * -12)
= 60 + (-120) + 240
= 180
Therefore, the dot product of F1 and F2 is 180.
To find the angle between two vectors, you can use the dot product formula and the magnitude of the vectors. The angle is given by the formula:
θ = cos^(-1)((F1 ⋅ F2) / (|F1| * |F2|))
Here, |F1| represents the magnitude of vector F1, and |F2| represents the magnitude of vector F2.
Let's calculate the angle between the given vectors:
|F1| = √(F1x^2 + F1y^2 + F1z^2) = √(10^2 + (-15)^2 + (-20)^2) = √(100 + 225 + 400) = √725 ≈ 26.92
|F2| = √(F2x^2 + F2y^2 + F2z^2) = √(6^2 + 8^2 + (-12)^2) = √(36 + 64 + 144) = √244 ≈ 15.62
θ = cos^(-1)((F1 ⋅ F2) / (|F1| * |F2|))
= cos^(-1)(180 / (26.92 * 15.62))
≈ cos^(-1)(180 / 421.19)
≈ cos^(-1)(0.427)
Using a calculator or a table of inverse cosine values, the angle is approximately 65.11 degrees.
Therefore, the dot product of F1 and F2 is 180, and the angle between them is approximately 65.11 degrees.