Let $$ P, Q $$, and $$ R $$ be points on a plane given by $$ P=(-1,-4,-3) $$, $$ Q=(2,4,3) $$, and $$ R=(-2,1,0) $$. Find the cross product $$ P Q \times P R $$. $$ P Q \times P R=[\text { Ex: } 42, \square, 23] $$

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UpStudy AI · Accepted Answer

We start by computing the vectors from point $$ P $$ to points $$ Q $$ and $$ R $$.

1. Compute the vector
   $$
   PQ = Q-P = \bigl(2-(-1),\, 4-(-4),\, 3-(-3)\bigr) = (3,8,6).
   $$

2. Compute the vector
   $$
   PR = R-P = \bigl((-2)-(-1),\, 1-(-4),\, 0-(-3)\bigr) = (-1,5,3).
   $$

3. The cross product $$ PQ \times PR $$ is given by:
   $$
   PQ \times PR = \left( 
   8\cdot3 - 6\cdot5,\quad 
   6\cdot(-1) - 3\cdot3,\quad 
   3\cdot5 - 8\cdot(-1) 
   \right).
   $$
   Now, compute each component:
   - $$x$$-component:
     $$
     8\cdot3 - 6\cdot5 = 24 - 30 = -6.
     $$
   - $$y$$-component:
     $$
     6\cdot(-1) - 3\cdot3 = -6 - 9 = -15.
     $$
   - $$z$$-component:
     $$
     3\cdot5 - 8\cdot(-1) = 15 + 8 = 23.
     $$

Thus, the cross product is
$$
PQ \times PR = [-6,\,-15,\,23].
$$
The cross product $$ PQ \times PR $$ is $$[-6, -15, 23]$$.

Let , and be points on a plane given by ,
, and .
Find the cross product .

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