The position vector of the centre of the circle |𝐫|=5, 𝐫 (𝐢+𝐣+𝐤)=3 √(3) is (A) √(3)(𝐢+𝐣+𝐤) (B) 𝐢

Name: Problem 6, Chapter 22: A Complete Resource Book in Mathematics for JEE Main
Uploaded: 2021-10-19T14:13:33-08:00
Duration: 3 min 13 s
Channel: Varsha Aggarwal
Description: The position vector of the centre of the circle |𝐫|=5, 𝐫 (𝐢+𝐣+𝐤)=3 √(3) is (A) √(3)(𝐢+𝐣+𝐤) (B) 𝐢+𝐣+𝐤 (C) 3(𝐢+𝐢+𝐤) (D) none of the above

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The position vector of the centre of the circle |𝐫|=5, 𝐫...

The position vector of the centre of the circle $|\mathbf{r}|=5, \mathbf{r}$ $(\mathbf{i}+\mathbf{j}+\mathbf{k})=3 \sqrt{3}$ is
(A) $\sqrt{3}(\mathbf{i}+\mathbf{j}+\mathbf{k})$
(B) $\mathbf{i}+\mathbf{j}+\mathbf{k}$
(C) $3(\mathbf{i}+\mathbf{i}+\mathbf{k})$
(D) none of the above

Key Concepts

Intersection of a Sphere and a Plane

When a plane cuts through a sphere, the intersection is a circle (if the plane passes through the sphere) whose properties depend on the relative position of the plane and the sphere’s centre. Understanding this intersection involves combining the sphere's equation and the plane's equation to analyze the resulting circle's centre and radius.

Position Vector

A position vector is a vector that extends from a fixed origin to a specific point in space. It is used to represent the coordinates of a point and is fundamental in describing locations and displacements in vector geometry.

Dot Product

The dot product is an operation that multiplies two vectors to yield a scalar. It is crucial in defining geometric relationships such as the angle between vectors and is used in formulating linear equations of planes by relating a vector to its normal.

Plane Equation

A plane in three-dimensional space can be defined using the dot product by the equation r · n = d, where r is the position vector of a variable point on the plane, n is a fixed normal vector to the plane, and d is the distance from the origin scaled by the magnitude of n. This formulation is vital for identifying all points lying on a given plane.

Distance from a Point to a Plane

The distance from a point to a plane is given by the absolute value of the difference between the plane’s constant term and the dot product of the point's position vector with the plane's unit normal vector. This concept is important when determining how far off a point is from a plane, such as finding the centre of a circle formed by the intersection of a sphere and a plane.

Projection onto a Plane

Projection involves finding the point on a plane that is closest to a given point, often by moving along a direction perpendicular to the plane. This concept is applied to find the centre of the circle formed by the intersection of a sphere and a plane, by projecting the sphere’s centre onto the plane.

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