The vectors 2 𝐢̂+3 𝐣̂-2 𝐤̂, 5 𝐢̂+a 𝐣̂+𝐤̂ and -𝐢̂+2 𝐣̂+3 𝐤̂ are coplanar when ' a ' is (a) -9

Name: Problem 13, Chapter 3: Master Resource Book in JEE Main Physics
Uploaded: 2022-01-14T06:15:36-08:00
Duration: 41 s
Channel: Hast Aggarwal
Description: The vectors 2 𝐢̂+3 𝐣̂-2 𝐤̂, 5 𝐢̂+a 𝐣̂+𝐤̂ and -𝐢̂+2 𝐣̂+3 𝐤̂ are coplanar when ' a ' is (a) -9 (b) 9 (c) -18 (d) 18

step 1 So let's solve this question if the three vectors are co -pregn and the skelet triple product is

Question

The vectors $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \quad 5 \hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}} \quad$ and $-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ are coplanar when ' $a$ ' is (a) $-9$ (b) 9 (c) $-18$ (d) 18

   The vectors $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \quad 5 \hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}} \quad$ and
$-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ are coplanar when ' $a$ ' is
(a) $-9$
(b) 9
(c) $-18$
(d) 18

Master Resource Book in JEE Main Physics

D. B. Singh 1st Edition

Chapter 3, Problem 13

Instant Answer

Solved by Expert Hast Aggarwal

01/14/2022

Step 1

The scalar triple product of vectors $\vec{A}$, $\vec{B}$, and $\vec{C}$ is given by $\vec{A} \cdot (\vec{B} \times \vec{C})$. Show more…

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The vectors $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \quad 5 \hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}} \quad$ and $-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ are coplanar when ' $a$ ' is (a) $-9$ (b) 9 (c) $-18$ (d) 18

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Key Concepts

Coplanarity of Vectors

Coplanarity of vectors refers to a condition where three or more vectors lie within the same plane. In three-dimensional space, any set of vectors is coplanar if there exists a single plane that contains them all. This concept is crucial when analyzing geometric relationships and configurations in linear algebra and vector calculus.

Scalar Triple Product

The scalar triple product is a mathematical operation involving three vectors that provides a measure of the volume of the parallelepiped formed by them. It is computed as the dot product of one vector with the cross product of the other two. A key property of the scalar triple product is that it becomes zero if the vectors are coplanar, making it an essential tool for determining coplanarity in vector analysis.

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