5. Cross Product

Homework

1. Find the determinant \(\begin{vmatrix} 4 & -2 & 3 \\ 1 & 0 & 5 \\ -2 & 4 & 1 \end{vmatrix}\)

2. Find the cross product \(\vec u\times\vec v\) when \(\vec u=(2,-5,1)\) and \(\vec v=(3,-1,-4)\)

3. Let \(\vec u=(2,3,0)\), \(\vec v=(-4,2,1)\), and \(\vec w=(-2,4,2)\).

   1. Find \(\vec u\cdot\vec v\times\vec w\).

   2. Find \(\vec w\times\vec v\cdot\vec u\).

4. Find the area of the parallelogram with adjacent edges \(\vec u=(3,-2,4)\) and \(\vec v=(-1,2,-3)\).

5. Find the area of the triangle with adjacent edges \(\vec p=(3,-2,-3)\) and \(\vec q=(1,-1,-2)\).

6. Find the area of a parallelogram with adjacent edge vectors \(\vec a\) and \(\vec b\), if \(|\vec a|=4\), \(|\vec b|=2\sqrt{3}\), and the angle between them is \(\theta=60^\circ\).

7. A triangle has vertices \(P=(1,2,3)\), \(Q=(4,1,4)\) and \(R=(2,0,1)\). Find its area.

8. A parallelogram has vertices \(A=(1,3,2)\), \(B=(1,6,5)\), \(C=(3,8,5)\) and \(D=(3,5,2)\), traversed in that order. Find its area.

9. Find the volume of the parallelpiped with adjacent edges \(\vec a=\left\langle3,-4,2\right\rangle\), \(\vec b=\left\langle-2,1,4\right\rangle\), and \(\vec c=\left\langle-1,1,1\right\rangle\)

10. Find the volume of the parallelpiped with adjacent edges \(\overrightarrow{PQ}\), \(\overrightarrow{PR}\), and \(\overrightarrow{PS}\) where \(P=(1,1,1)\), \(Q=(3,2,4)\), \(R=(0,4,-1)\), and \(S=(2,-1,4)\).

11. A rigid object is held fixed at the point \(P=(1,2,3)\). Find the torque on the object, if the force \(\vec F=\langle3,-2,1\rangle\) is applied at the point \(Q=(2,3,4)\).

12. A triangular plate with vertices \(P=(-3,0,0)\), \(Q=(2,0,0)\) and \(R=(0,1,0)\) is held fixed at the origin \(O=(0,0,0)\). The force \(\vec F_1=\left\langle 1,2,0 \right\rangle\) is applied at \(P\) and the force \(\vec F_2=\left\langle -2,1,0\right\rangle\) is applied at \(Q\). What force \(\vec F_3=\left\langle a,b,c\right\rangle\) can be applied at \(R\) to keep the triangle from rotating?