Show that the four points A, B, C and D with position vectors 4^i+5^j+^k,−^j−^k,3^i+9^j+4^k and 4(−^i+^j+^k) respectively are coplannar.
OR
The scalar product of the vector →a=^i+^j+^k with a unit vector along the sum of vector →b=2^i+4^j+5^k and →c=λ^i+2^j+3^k is equal to one. Find the value of λ and hence find the unit vector along →b+→c.
Show that the four points A, B, C and D with position vectors 4^i+5^j+^k,−^j−^k,3^i+9^j+4^k and 4(−^i+^j+^k) respectively are coplannar.
OR
The scalar product of the vector →a=^i+^j+^k with a unit vector along the sum of vector →b=2^i+4^j+5^k and →c=λ^i+2^j+3^k is equal to one. Find the value of λ and hence find the unit vector along →b+→c.
The points A, B, C and D are coplanar if ¯AB¯ACׯAD=0
Now, ¯AB=−4^i−6^j−2^k,¯AC=−^i+4^j+3^k,¯AD=−8^i−^j+3^k
∴¯AB.¯ACׯAD=∣∣
∣∣−4−6−2−143−8−13∣∣
∣∣=−4(12+3)+6(−3+24)−2(1+32)=0
So, A, B, C and D are coplanar
OR
Given ¯a.→b+→c|→b+→c=1 ⇒→a.→b+→a.→c=|→b+→c|
⇒(^i+^j+^k).(2^i+4^j−5^k)+(^i+^j+^k).(λ^i+2^j+3^k)=|2^i+4^j−5^k+λ^i+2^j+3^k|
⇒2+4−5+λ+2+3=√(λ+2)2+62+(−2)2 ∴λ=1
Also, unit vector along →b+→c is given as:
→b+→c|→b+→c|=(λ+2)^i+6^j−2^k√(λ+2)2+62+(−2)2=(1+2)^i+6^j−2^k√(1+2)2+62+(−2)2=3^i+6^j−2^k7
The points A, B, C and D are coplanar if ¯AB¯ACׯAD=0
Now, ¯AB=−4^i−6^j−2^k,¯AC=−^i+4^j+3^k,¯AD=−8^i−^j+3^k
∴¯AB.¯ACׯAD=∣∣ ∣∣−4−6−2−143−8−13∣∣ ∣∣=−4(12+3)+6(−3+24)−2(1+32)=0
So, A, B, C and D are coplanar
OR
Given ¯a.→b+→c|→b+→c=1 ⇒→a.→b+→a.→c=|→b+→c|
⇒(^i+^j+^k).(2^i+4^j−5^k)+(^i+^j+^k).(λ^i+2^j+3^k)=|2^i+4^j−5^k+λ^i+2^j+3^k|
⇒2+4−5+λ+2+3=√(λ+2)2+62+(−2)2 ∴λ=1
Also, unit vector along →b+→c is given as:
→b+→c|→b+→c|=(λ+2)^i+6^j−2^k√(λ+2)2+62+(−2)2=(1+2)^i+6^j−2^k√(1+2)2+62+(−2)2=3^i+6^j−2^k7
