If the three vectors are coplanar, then value of ' x^' is: A=î-2 ĵ+3 k̂ B⃗=x ĵ+3 k̂ C⃗=7 î+3 -11

step 1 In this question it is given that three vectors are coplanar then we have to find the value of x

If the three vectors are coplanar, then value of ' x^' is:...

If the three vectors are coplanar, then value of ' $x^{\prime}$ is:
$\overrightarrow{\mathrm{A}}=\hat{i}-2 \hat{j}+3 \hat{\mathrm{k}}$
$\vec{B}=x \hat{j}+3 \hat{k}$
$\vec{C}=7 \hat{i}+3 \hat{\jmath}-11 \hat{k}$
(a) $36 / 21$
(b) $-51 / 32$
(c) $51 / 32$
(d) $-36 / 21$

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Key Concepts

Coplanarity of Vectors

Coplanarity refers to the condition where multiple vectors lie on the same geometric plane. For three vectors in three-dimensional space, they are coplanar if one of them can be expressed as a linear combination of the other two, meaning they do not span the entire 3D space.

Scalar Triple Product

The scalar triple product is a key tool to check for coplanarity. It is computed as the dot product of one vector with the cross product of the other two. If the resulting scalar value is zero, it indicates that the vectors lie in the same plane.

Determinant

The determinant of a 3x3 matrix formed by the components of the three vectors is equivalent to the scalar triple product. A zero determinant confirms that the vectors are dependent and lie on a common plane.

Vector Operations

Vector operations like the cross product and dot product are fundamental in evaluating the relationships between vectors. The cross product gives a perpendicular vector, and its dot product with another vector helps in determining geometric properties such as coplanarity.

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