Question

.. If A(5,3), B(0,0) & C(-3,5) are the vertices of the triangle, then value of AB vector .BC vector is :​

Answer 1

Answer:

The vertices of ΔABC are given as A(1,2,3),B(−1,0,0), and C(0,1,2).

Also, it is given that ∠ABC is the angle between the vectors

BA

and

BC

.

BA

={1−(−1)}

i

^

+(2−0)

j

^

+(3−0)

k

^

=2

i

^

+2

j

^

+3

k

^

BC

={0−(−1)}

i

^

+(1−0)

j

^

+(2−0)

k

^

=

i

^

+

j

^

+2

k

^

∴

BA

⋅

BC

=(2

i

^

+2

j

^

+3

k

^

)⋅(

i

^

+

j

^

+2

k

^

)=2×1+2×1+3×2=2+2+6=10

∣

BA

∣=

2

2

+2

2

+3

2

=

4+4+9

=

17

∣

BC

∣=

1+1+2

2

=

6

Now, it is known that:

BA

⋅

BC

=∣

BA

∣∣

BC

∣cos(∠ABC)

∴10=

17

×

6

cos(∠ABC)

⇒cos(∠ABC)=

17

×

6

10

⇒∠ABC=cos

−1

(

102

10

)