Show that the vectors a → , b → , c → are coplanar, when a → = 2 i ^ - j ^ + 2 k ^ , b → = i ^ + 2 j ^ - 3 k ^ and c → = 3 i ^ - 4 j ^ + 7 k ^

Given a → = 2 i ^ - j ^ + 2 k ^ , b → = i ^ + 2 j ^ - 3 k ^ and c → = 3 i ^ - 4 j ^ + 7 k ^ The given vectors will be coplanar if a → b → c → = 0 . a → b → c → = 2 - 1 2 1 2 - 3 3 - 4 7 2 - 1 2 1 2 - 3 3 - 4 7 = 2 2 - 3 - 4 7 + 1 1 - 3 3 7 + 2 1 2 3 - 4 = 2 ( 14 - 12 ) + 1 ( 7 + 9 ) + 2 ( - 4 - 6 ) = 2 ( 2 ) + 1 ( 16 ) - 2 ( 10 ) = 4 + 16 - 20 = 0 Hence three vectors are coplanar.

Show that the vectors a → , b → , c → are coplanar, when a → = 2 i ^ - j ^ + 2 k ^ , b → = i ^ + 2 j ^ - 3 k ^ and c → = 3 i ^ - 4 j ^ + 7 k ^

Given a → = 2 i ^ - j ^ + 2 k ^ , b → = i ^ + 2 j ^ - 3 k ^ and c → = 3 i ^ - 4 j ^ + 7 k ^ The given vectors will be coplanar if a → b → c → = 0 . a → b → c → = 2 - 1 2 1 2 - 3 3 - 4 7 2 - 1 2 1 2 - 3 3 - 4 7 = 2 2 - 3 - 4 7 + 1 1 - 3 3 7 + 2 1 2 3 - 4 = 2 ( 14 - 12 ) + 1 ( 7 + 9 ) + 2 ( - 4 - 6 ) = 2 ( 2 ) + 1 ( 16 ) - 2 ( 10 ) = 4 + 16 - 20 = 0 Hence three vectors are coplanar.

Show that the vectors a → = ( i ^ + 3 j ^ + k ^ ) , b → = ( 2 i ^ - j ^ - k ^ ) and c → = ( 7 j ^ + 3 k ^ ) are parallel to the same plane.

Given vectors a → = ( i ^ + 3 j ^ + k ^ ) , b → = ( 2 i ^ - j ^ - k ^ ) and c → = ( 7 j ^ + 3 k ^ ) If three vectors are parallel to the same plane, then a → b → c → = 0 a → b → c → = 1 3 1 2 - 1 - 1 0 7 3 ⇒ 1 3 1 2 - 1 - 1 0 7 3 = 1 - 1 - 1 7 3 - 3 2 - 1 0 3 + 1 2 - 1 0 7 = 1 ( - 3 + 7 ) - 3 6 - 0 + 1 14 - 0 = 1 ( 4 ) - 3 ( 6 ) + 1 ( 14 ) = 4 - 18 + 14 = 0 This means a → b → c → are parallel to the same plane.

If the vectors ( a i ^ + a j ^ + c k ^ ) , ( i ^ + k ^ ) and ( c i ^ + c j ^ + b k ^ ) be coplanar, show that c 2 = a b .

Given vectors ( a i ^ + a j ^ + c k ^ ) , ( i ^ + k ^ ) and ( c i ^ + c j ^ + b k ^ ) If three vectors a → , b → , c → are coplanar, then their scalar triple product is equal to zero. ⇒ [ a → b → c → ] = 0 ⇒ a a c 1 0 1 c c b = 0 ⇒ a 0 1 c b - a 1 1 c b + c 1 0 c c = 0 ⇒ a ( - c ) - a ( b - c ) + c c = 0 ⇒ - a c - a b + a c + c 2 = 0 ⇒ - a b + c 2 = 0 ⇒ c 2 = a b Hence proved

Show that the four points with position vectors ( 4 i ^ + 8 j ^ + 12 k ^ ) , ( 2 i ^ + 4 j ^ + 6 k ^ ) , ( 3 i ^ + 5 j ^ + 4 k ^ ) and ( 5 i ^ + 8 j ^ + 5 k ^ ) are coplanar.

Given vectors O A → = 4 i ^ + 8 j ^ + 12 k ^ O B → = 2 i ^ + 4 j ^ + 6 k ^ O C → = 3 i ^ + 5 j ^ + 4 k ^ O D → = 5 i ^ + 8 j ^ + 5 k ^ a → = A B → = O B → - O A → = - 2 i ^ - 4 j ^ - 6 k ^ b → = B C → = O C → - O B → = i ^ + j ^ - 2 k ^ c → = C D → = O D → - O C → = 2 i ^ + 3 j ^ + k ^ The given vectors will be coplanar if a → b → c → = 0 . a → b → c → = - 2 - 4 - 6 1 1 - 2 2 3 1 = - 2 1 - 2 3 1 + 4 1 - 2 2 1 - 6 1 1 2 3 = - 2 1 + 6 + 4 ( 1 + 4 ) - 6 ( 3 - 2 ) = - 2 ( 7 ) + 4 ( 5 ) - 6 ( 1 ) = - 14 + 20 - 6 = 0 since a → , b → , c → are coplanar, this means the given points are coplanar.

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 4

MEDIUM

12th CBSE

IMPORTANT

Earn 100

Show that the vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar, when $\vec{a} = 2 \hat{i} - \hat{j} + 2 \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - 3 \hat{k} and \vec{c} = 3 \hat{i} - 4 \hat{j} + 7 \hat{k}$

Important Questions on Product of Three Vectors

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 5

Find the value of $λ$ for which the vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar, when $\vec{a} = (2 \hat{i} - \hat{j} + \hat{k}), \vec{b} = (\hat{i} + 2 \hat{j} + 3 \hat{k}) and \vec{c} = (3 \hat{i} + λ \hat{j} + 5 \hat{k})$ .

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 5

Find the value of $λ$ for which the vectors, $\vec{a}, \vec{b}, \vec{c}$ are coplanar, when $\vec{a} = λ \hat{i} - 10 \hat{j} - 5 \hat{k}, \vec{b} = - 7 \hat{i} - 5 \hat{j} and \vec{c} = \hat{i} - 4 \hat{j} - 3 \hat{k}$ .

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 5

Find the value of $λ$ for which the vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar, when $\vec{a} = \hat{i} - \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} + \hat{j} - \hat{k} and \vec{c} = λ \hat{i} - \hat{j} + λ \hat{k}$

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 6

If $\vec{a} = (2 \hat{i} - \hat{j} + \hat{k}), \vec{b} = (\hat{i} - 3 \hat{j} - 5 \hat{k})$ and $\vec{c} = (3 \hat{i} - 4 \hat{j} - \hat{k}),$ find $[\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]$ in cubic units and interpret the result.

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 7

The volume of the parallelepiped whose edges are $(- 12 \hat{i} + λ \hat{k}), (3 \hat{j} - \hat{k}) and (2 \hat{i} + \hat{j} - 15 \hat{k})$ is $546$ cubic units. Find the value of $λ$ .

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 8

Show that the vectors $\vec{a} = (\hat{i} + 3 \hat{j} + \hat{k}), \vec{b} = (2 \hat{i} - \hat{j} - \hat{k}) and \vec{c} = (7 \hat{j} + 3 \hat{k})$ are parallel to the same plane.

MEDIUM

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 9

If the vectors $(a \hat{i} + a \hat{j} + c \hat{k}), (\hat{i} + \hat{k}) and (c \hat{i} + c \hat{j} + b \hat{k})$ be coplanar, show that $c^{2} = a b$ .

HARD

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 25 - Product of Three Vectors>EXERCISE>Q 10

Show that the four points with position vectors $(4 \hat{i} + 8 \hat{j} + 12 \hat{k}), (2 \hat{i} + 4 \hat{j} + 6 \hat{k}), (3 \hat{i} + 5 \hat{j} + 4 \hat{k})$ and $(5 \hat{i} + 8 \hat{j} + 5 \hat{k})$ are coplanar.