Find the Norm u*(v[[i,j,k],[3,1,0],[2,0,4]])

Problem

∥u×v∥,u=<3,1,0>,v=<2,0,4>

Solution

1. Identify the vectors u and v from the given components: u=<3,1,0> and v=<2,0,4>

2. Calculate the cross product u×v using the determinant of a 3×3 matrix.

u×v=|[i,j,k],[3,1,0],[2,0,4]|

3. Expand the determinant along the first row.

u×v=i*(1⋅4−0⋅0)−j*(3⋅4−0⋅2)+k*(3⋅0−1⋅2)

4. Simplify the components of the resulting vector.

u×v=4*i−12*j−2*k=<4,−12,−2>

5. Apply the norm formula ∥w∥=√(,x2+y2+z2) to the cross product vector.

∥u×v∥=√(,4+(−12)2+(−2)2)

6. Evaluate the squares and the sum inside the square root.

∥u×v∥=√(,16+144+4)

∥u×v∥=√(,164)

7. Simplify the radical by factoring out the largest perfect square.

∥u×v∥=√(,4⋅41)

∥u×v∥=2√(,41)

Final Answer

∥u×v∥=2√(,41)

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