If a- b- c- are the vertices of the triangle ABC then a-b-b-c-c-a-

The correct option is D Normal Vector to the plane containing a⃗,b⃗,c⃗ The sides of the triangle is given by, A⃗B⃗=b⃗ a⃗, and B⃗C⃗=c⃗ b⃗ Then ar ...

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Standard XII

Mathematics

Applications of Cross Product

If a⃗, b⃗, ...

Question

If $\to a, \to b, \to c$ are the vertices of the triangle $A B C$ then $\to a \times \to b + \to b \times \to c + \to c \times \to a =$

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Area of the triangle

A B C

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Normal Vector to the plane containing

\to a, \to b, \to c

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Solution

The correct option is D Normal Vector to the plane containing $\to a, \to b, \to c$
The sides of the triangle is given by,
$\to A B = \to b - \to a,$ and $\to B C = \to c - \to b$
Then area of the triangle is
$= \frac{1}{2} \to A B \times \to B C$
$= \frac{1}{2} (\to b - \to a) \times (\to c - \to b)$
$= \frac{1}{2} [\to b \times (\to c - \to b) - \to a (\to c - \to b)]$
$= \frac{1}{2} [(\to b \times \to c) - (\to b \times \to b) - (\to a \times \to c) + (\to a \times \to b)]$
Now $\to b \times \to b = 0$
Hence, $= \frac{1}{2} [(\to b \times \to c) - (\to b \times \to b) - (\to a \times \to c) + (\to a \times \to b)]$
$= \frac{1}{2} [(\to b \times \to c) - (\to a \times \to c) + (\to a \times \to b)]$
$= \frac{1}{2} [(\to b \times \to c) + (\to c \times \to a) + (\to a \times \to b)]$
$\frac{1}{2} \to A B \times \to B C = \frac{1}{2} [(\to b \times \to c) + (\to c \times \to a) + (\to a \times \to b)]$
$\to A B \times \to B C = (\to b \times \to c) + (\to c \times \to a) + (\to a \times \to b)$
But $\to A B \times \to B C$ will give a vector perpendicular to the plane containing $A B$ and $B C$ , that is perpendicular to the plane of the triangle $A B C$ .
Hence, $(\to b \times \to c) + (\to c \times \to a) + (\to a \times \to b)$ is vector parallel to the normal vector of the plane of triangle $A B C$ .

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Similar questions

Q. Let

\to A = \to b \times \to c, \to B = \to c \times \to a, \to C = \to a \times \to b

, then the vectors

\to A \times (\to B \times \to C)

\to B \times (\to C \times \to A)

, and

\to C \times (\to A \times \to B)

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If $\to a, \to b a n d \to c$ determine the vertices of a triangle, show that $\frac{1}{2} [\to b \times \to c + \to c \times \to a + \to a \times \to b]$ gives the vector area of the triangle. Hence, deduce the condition that the three points $\to a, \to b a n d \to c$ are collinear. Also, find the vector normal to the plane of the triangle.