CameraIcon

SearchIcon

MyQuestionIcon

1

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

Applications of Cross Product

If A, B, C ar...

Question

If A, B, C are three non-collinear points with position vectors $\to a, \to b, \to c$ , respectively, then show that the length of the perpendicular from C on AB is $\frac{| (\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) |}{| \to b - \to a |}$ .

Open in App

Solution

Consider $△ A B C$ , with base AB.
So, length of the perpendicular from C on AB is the height of the triangle, $h$ .
Method 1:
$h = A C sin θ$ , where $θ$ is the angle between AB and AC.
$\Rightarrow h = \frac{| \to A B | | \to A C | sin θ}{| \to A B |} = \frac{| \to A B \times \to A C |}{| \to A B |} = \frac{| (\to b - \to a) \times (\to c - \to a) |}{| \to b - \to a |} = \frac{| (\to b \times \to c) - (\to a \times \to c) - (\to b \times \to a) |}{| \to b - \to a |}$
$\Rightarrow h = \frac{| (\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) |}{| \to b - \to a |}$
Method 2:
$h = \frac{A r e a o f T r i a n g l e}{\frac{1}{2} A B} = \frac{\frac{1}{2} | \to A B \times \to A C |}{\frac{1}{2} | \to A B |}$
$= \frac{| \to A B \times \to A C |}{| \to A B |} = \frac{| (\to b - \to a) \times (\to c - \to a) |}{| \to b - \to a |} = \frac{| (\to b \times \to c) - (\to a \times \to c) - (\to b \times \to a) |}{| \to b - \to a |}$
$\Rightarrow h = \frac{| (\to a \times \to b) + (\to b \times \to c) + (\to c \times \to a) |}{| \to b - \to a |}$
Hence, proved.

Suggest Corrections

thumbs-up

0

Similar questions

Q. The length of the perpendicular from the origin to the plane passing through three non-collinear points

\to a, \to b, \to c

Q. The length of the perpendicular from the origin to the plane passing through three non-collinear points

\to a, \to b, \to c

\to a, \to b, \to c

are non-zero non-collinear vectors such that

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

\to a + \to b + \to c =

Q. Prove that points

A, B, C

having positions vectors

\to a, \to b, \to c

are collinear, if

[\to b \times \to c + \to c \times \to a + \to a \times \to b] = \to 0

\to a, \to b, \to c

are positive vectors of vertices

A, B, C

Δ A B C

\to r

is position vector of a point

P

(| \to b - \to c | + | \to c - \to a | + | \to a - \to b |) \to r = | \to b - \to c | \to a + | \to c - \to a | \to b + ∣ ∣ \to a - \to b ∣ ∣ \to c

P

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Applications of Cross Product

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Applications of Cross Product

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon