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Vector Addition

If a⃗, b⃗ ,...

Question

If $\to a, \to b, \to c$ are position vectors of the vertices of a triangle ABC, then length of the perpendicular drawn from C to AB is

A

\frac{∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣}{∣ ∣ \to a - \to b ∣ ∣}

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B

\frac{∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣}{∣ ∣ \to b - \to c ∣ ∣}

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C

\frac{∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣}{| \to a - \to c |}

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D

∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣

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Solution

The correct option is A $\frac{∣ ∣ \to a \times \to b + \to b \times \to c + \to c \times \to a ∣ ∣}{∣ ∣ \to a - \to b ∣ ∣}$

$Solve:-$
$Let A B C is a given triangle$
$having position vector of vertices be \to a,$
$\to b and \to c respectively$
$Area of triangle = \frac{1}{2} (A B) (C P)$

$= \frac{1}{2} (A B) (A C) sin A$

$\Rightarrow - - \to A C \times - - \to A B = | \to a \times \to b + \to b \times \to c + \to c \times \to a |$

$\Rightarrow | A B | | A C | sin A = | \to a \times \to b + \to b \times \to c + \to c \times \to a |$

$\Rightarrow | \to b - a | | C P | = | \to a \times \to b + \to b \times \to c + \to c \times \to a |$

$\Rightarrow | C P | = | A C | sin A = \frac{| \to a \times \to b + \to b \times \to c + \to c \times \to a |}{| \to b - \to a |}$

$\Rightarrow | C P | = \frac{| \to a \times \to b + \to b \times \to c + \to c \times \to a |}{| \to a - ¯ b |}$

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Similar questions

\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)

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

\to a, \to b, \to c

are three vectors such that

\to a \times \to b = \to c, \to b \times \to c = \to a, \to c \times \to a = \to b

then prove that

| \to a | = | \to b | = | \to c |

(\to a + \to b) \times (\to a - \to b) + (\to b + \to c) \times (\to b - \to c) + (\to c + \to a) \times (\to c - \to a) =

\to a =^i +^j +^k

\to b = - 3^i +^k

\to c =^i + 2^j + 3^k

, then the ascending order of magnitudes of

A)

[\to a \times \to b

\to b \times \to c

\to c \times \to a]

B)

[\to a + \to b

\to b + \to c

\to c + \to a]

C)

[\to a

\to b

\to c]

[\to a

\to b

\to c]^{- 1}

D)

[\to a - \to b

\to b - \to c

\to c - \to a]

\to a, \to b

\to c

are perpendicular to

\to b + \to c, \to c + \to a

\to a + \to b

respectively and if

| \to a + \to b | = 6, | \to b + \to c | = 8

| \to c + \to a | = 10

| \to a + \to b + \to c |

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