Let a- b- c be non coplanar vectors and p-b-c- a b c - q-c-a- b c a - r-a-b- c a b - then

The correct options are A p·a=1 C p·a+q·b+r·c=3 p = b × c #160; a .( b × c #160;) #160; , q = c × a #160; b .( c × a #160;) ...

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Byju's Answer

Standard XII

Mathematics

Test for Coplanarity

Let a, b,...

Question

Let $\to a$ , $\to b$ , $\to c$ be non coplanar vectors and $\to p = \frac{\to b \times \to c}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}$ , $\to q = \frac{\to c \times \to a}{[\begin{matrix} \to b & \to c & \to a \end{matrix}]}$ , $\to r = \frac{\to a \times \to b}{[\begin{matrix} \to c & \to a & \to b \end{matrix}]}$ , then

\to p \cdot \to a = 1

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\to p \cdot \to a + \to q \cdot \to b + \to r \cdot \to c = 3

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\to p \cdot \to a + \to q \cdot \to b + \to r \cdot \to c = 0

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None of these

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Solution

The correct options are
A $\to p \cdot \to a = 1$
C $\to p \cdot \to a + \to q \cdot \to b + \to r \cdot \to c = 3$
$\to p = \frac{\to b \times \to c}{\to a . (\to b \times \to c)}, \to q = \frac{\to c \times \to a}{\to b . (\to c \times \to a)}, \to r = \frac{\to a \times \to b}{\to c . (\to a \times \to b)}$
$\to p . \to a = \frac{\to a . (\to b \times \to c)}{\to a . (\to b \times \to c)} = 1$ Similarly $\to q . \to b = \to r . \to c = 1$
$\to p . \to a + \to q . \to b + \to r . \to c = 3$
A and B are right.

Suggest Corrections

Similar questions

Q. If

\to a, \to b

and

\to c

are three non-coplanar vectors and

\to p, \to q

and

\to r

are vectors defined by

\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}

and

\to r = \frac{\to a \times \to b}{[\to a \to b \to c]}

, then the value of

(\to a + \to b) \cdot \to p + (\to b + \to c) \cdot \to q + (\to c + \to a) \cdot \to r =

Q. If

∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

and vectors

(1, a, a^{2}), (1, b, b^{2})

and

(1, c, c^{2})

are non coplanar then

a b c

equals

Q. If

∣ ∣ ∣ ∣ ∣ \begin{matrix} a & a^{2} & 1 + a^{3} b & b^{2} & 1 + b^{3} c & c^{2} & 1 + c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

and the vectors

\to A = 1, a, a^{2}, \to B = 1, b, b^{2}, \to C = 1, c, c^{2}

are non-coplanar, then find the value of product

a b c

Q. If

¯ a, ¯ b, ¯ c

are non coplanar vectors and

λ \in R

then

¯ a + 2 ¯ b + 3 ¯ c, λ ¯ b + 4 ¯ c

and

(2 λ - 1) ¯ c

are non coplanar for

Q. Let

a, b

and

c

be three non-coplanar vectors, and let

p, q

and

r

be the vectors defined by the relation

p = \frac{b \times c}{[a b c]}, q = \frac{c \times a}{[a b c]},

and

r = \frac{a \times b}{[a b c]},

Then the value of the expression

(a + b) \cdot p + (b + c) \cdot q + (c + a) \cdot r

is equal to

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Let →a, →b, →c be non coplanar vectors and →p=→b×→c[→a→b→c], →q=→c×→a[→b→c→a], →r=→a×→b[→c→a→b], then