CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Test for Coplanarity

If a,b,c are ...

Question

If $\to a, \to b, \to c$ are three non coplanar vectors and $\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}, \to r = \frac{\to a \times \to b}{[\to a \to b \to c]},$ then $(2 \to a + 3 \to b + 4 \to c) \cdot \to p + (2 \to b + 3 \to c + 4 \to a) \cdot \to q + (2 \to c + 3 \to a + 4 \to b) \cdot \to r =$

Open in App

Solution

Given : $\to p = \frac{\to b \times \to c}{[\to a \to b \to c]}, \to q = \frac{\to c \times \to a}{[\to a \to b \to c]}, \to r = \frac{\to a \times \to b}{[\to a \to b \to c]},$
$\Rightarrow \to a \cdot \to p = \frac{\to a \cdot (\to b \times \to c)}{[\to a \to b \to c]} = 1, \to b \cdot \to p = \frac{\to b \cdot (\to b \times \to c)}{[\to a \to b \to c]} = 0, \to c \cdot \to p = \frac{\to c \cdot (\to b \times \to c)}{[\to a \to b \to c]} = 0$
( $∵$ scalar triple product is zero if two vectors are same.)
similarly,
$\to a \cdot \to q = \to c \cdot \to q = 0, \to b \cdot \to q = 1$ and
$\to a \cdot \to r = \to b \cdot \to r = 0, \to c \cdot \to r = 1$
Now,
$(2 \to a + 3 \to b + 4 \to c) \cdot \to p + (2 \to b + 3 \to c + 4 \to a) \cdot \to q + (2 \to c + 3 \to a + 4 \to b) \cdot \to r = (2 + 0 + 0) + (2 + 0 + 0) + (2 + 0 + 0) = 6$

Suggest Corrections

thumbs-up

2

Similar questions

\to a, \to b

\to c

are non-coplanar, then

\frac{[\to a + 2 \to b \to b + 2 \to c \to c + 2 \to a]}{[\to a \to b \to c]} =

\to a, \to b

\to c

are three non-coplanar vectors and

\to p, \to q

\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]}

\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 =

\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]}

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

\to a, \to b

\to c

are three non-coplanar vectors, then the value of the expression

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

\to a, \to b, \to c

are three non-coplanar vectors,

\to p, \to q, \to r

are non-zero vectors such that

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

then the value of the expression

(\to a + \to b + \to c), (\to p + \to q + \to r)

\to a, \to b, \to c

are noncoplanar vectors and

\to p, \to p, \to r

\to p = \frac{\to b \times \to c}{[\to b \to c \to a]}, \to q = \frac{\to c \times \to a}{[\to c \to a \to b]}, \to r = \frac{\to a \times \to b}{[\to a \to b \to c]}, (\to a + \to b) . \to p + (\to b + \to c) . \to q + (\to c + \to a) . \to r

View More

Join BYJU'S Learning Program

explore_more_icon

Explore more

Test for Coplanarity

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon