Question

Given the vectors P=4i – 2j +3k, Q=2i +4j – 5K and S=S xi – j +2k, determine the value of S x for which the three vectors are coplanar.

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Answer to a math question Given the vectors P=4i – 2j +3k, Q=2i +4j – 5K and S=S xi – j +2k, determine the value of S x for which the three vectors are coplanar.

Expert avatar
Fred
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113 Answers
Para que los tres vectores sean coplanares, el determinante formado por los tres vectores debe ser igual a cero.

Se tiene que los vectores son:
P = 4\textbf{i} - 2\textbf{j} + 3\textbf{k} , Q = 2\textbf{i} + 4\textbf{j} - 5\textbf{k} y S = S_x\textbf{i} - \textbf{j} + 2\textbf{k} .

Ahora formamos el determinante con estos tres vectores y lo igualamos a cero:
\begin{vmatrix}4 & -2 & 3 \2 & 4 & -5 \S_x & -1 & 2\end{vmatrix}= 0

Expandiendo el determinante con la regla de Sarrus, tenemos:
4(4 \cdot 2 - (-5) \cdot (-1)) - (-2)(2 \cdot 2 - (-5) \cdot S_x) + 3(2 \cdot (-1) - 4S_x) = 0
4(8 - 5) + 4 + 3(-2 - 4S_x) = 0
12 - 4 - 12 - 12S_x = 0
-4 - 12 - 12S_x = 0
-16 -12S_x = 0
12S_x = -16
S_x = \frac{-16}{12}
\boxed{S_x = -\frac{4}{3}}

Por lo tanto, el valor de S_x para que los tres vectores sean coplanares es -\frac{4}{3} .

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