1.

If the volume of a parallelepiped whose coterminous edges are given by the vectors vec a=hat i+hat j+nhat k vec b=2i+4j-nhat k and vec c=hat i+hat j+3hat k(n>=0) is 158 cu.Units then​

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Let the VOLUME of a parallelopiped whose coterminous edges are given by

u

=

i

^

+

j

^

+λ

K

^

,

v

=

i

^

+

j

^

+3

k

^

and

w

=2

i

^

+

j

^

+

k

^

be 1cu. UNITS. If θ be the angle between the edge

u

and

w

, then cosθ can be:

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Answer

Correct option is

D

6

3

7

Given,

u

=

i

^

+

j

^

+λ

k

^

,

v

=

i

^

+

j

^

+3

k

^

w

=2

i

^

+

j

^

+

k

^

Volume of parallelopied =u.(

v

×

w

)=[

u

v

w

]

=

∣

∣

∣

∣

∣

∣

∣

∣

1

1

2

1

1

1

λ

λ

1

∣

∣

∣

∣

∣

∣

∣

∣

=1(1−3)−1(1−6)+λ(1−2)−2+5+(−λ)3−λ

so,

Given volume =±1 taking (+) sign

3−λ=1

λ=2

Angle between

u

and

w

cosθ=

∣

u

∣.∣

w

∣

u

.

w

=

1+1+4

.

4+1+1

(i+j+2k).(2i+j+k)

=

5

.

5

2+1+2

=

5

5

=1

Taking (−) sign

3−λ=−1

λ=4

u

=

i

^

+

j

^

+4

k

^

w

=2i+j+

k

^

cosθ=

∣

u

∣∣

w

∣

u

.

w

=

1+1+16

4+1+1

(

i

^

+

j

^

+4

k

^

).(2i+j+k)

=

18

6

2+1+4

=

6

3

....Answer


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