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Check that the four points $P(2,4,4)$ , $Q(3,1,6)$ , $R(2,8,0)$ , and $S(5,2,3)$ all lie in a plane

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Estimate positive and negative square roots

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Q. Check that the four points

P(2,4,4)

,

Q(3,1,6)

,

R(2,8,0)

, and

S(5,2,3)

all lie in a plane

Calculate Vectors: Find the vectors $PQ$ , $PR$ , and $PS$ using the coordinates of the points. $\newline$ $PQ = Q - P = (3 - 2, 1 - 4, 6 - 4) = (1, -3, 2)$ $\newline$ $PR = R - P = (2 - 2, 8 - 4, 0 - 4) = (0, 4, -4)$ $\newline$ $PS = S - P = (5 - 2, 2 - 4, 3 - 4) = (3, -2, -1)$
Check Coplanarity: Check if the vectors PQ, PR, and PS are coplanar by finding the scalar triple product. $\newline$ The scalar triple product of vectors $a$ , $b$ , and $c$ is given by $[a b c] = a \cdot (b \times c)$ . $\newline$ First, calculate the cross product of PR and PS. $\newline$ PR \times PS = \left| \begin{array}{ccc}\(\newlinei & j & k (\newline\)0 & 4 & -4 (\newline\)3 & -2 & -1 (\newline\)\end{array} \right|\) $\newline$ $= (4 \cdot -1 - (-4) \cdot -2)i - (0 \cdot -1 - (-4) \cdot 3)j + (0 \cdot -2 - 4 \cdot 3)k$ $\newline$ $= (-4 - 8)i - (0 + 12)j + (0 - 12)k$ $\newline$ $= (-12)i - (12)j - (12)k$ $\newline$ $= (-12, -12, -12)$
Calculate Dot Product: Now, calculate the dot product of $PQ$ and the cross product of $PR$ and $PS$ . $PQ \cdot (PR \times PS) = (1, -3, 2) \cdot (-12, -12, -12)$ = $1*(-12) + (-3)*(-12) + 2*(-12)$ = $-12 + 36 - 24$ = $0$

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