Let and be three non-zero non-coplanar vectors. Let the position vector of four points A, B, C and D be and respectively. If and are coplanar, then is equal to ________ . [2023]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vector of four points A, B, C and D be $\vec{a} - \vec{b} + \vec{c}, λ \vec{a} - 3 \vec{b} + 4 \vec{c}, - \vec{a} + 2 \vec{b} - 3 \vec{c}$ and $2 \vec{a} - 4 \vec{b} + 6 \vec{c}$ respectively. If $\vec{A B}, \vec{A C}$ and $\vec{A D}$ are coplanar, then $λ$ is equal to ________ . [2023]

Ans.
(2)

$\vec{A B} = \vec{B} - \vec{A} = (λ - 1) \vec{a} - 2 \vec{b} + 3 \vec{c}$

$\vec{A C} = \vec{C} - \vec{A} = - 2 \vec{a} + 3 \vec{b} - 4 \vec{c}, \vec{A D} = \vec{a} - 3 \vec{b} + 5 \vec{c}$

$Since \vec{A B}, \vec{A C}, \vec{A D} are coplanar,$

So, $| \begin{matrix} λ - 1 & - 2 & 3 \\ - 2 & 3 & - 4 \\ 1 & - 3 & 5 \end{matrix} |$ $= 0$

$\Rightarrow (λ - 1) (15 - 12) + 2 (- 10 + 4) + 3 (6 - 3) = 0$

$\Rightarrow$ $λ = 2$

Want Us to Email you About Special Offers & Updates?