Condition of Tangency (Vector Form)

In 3D geometry, the condition of tangency refers to the specific criteria that must be met for a line to be tangent to a surface, such as a sphere or a plane. When dealing with vectors, these conditions can be expressed in a form that utilizes vector notation and operations. Understanding the condition of tangency is crucial for solving problems related to tangents in vector calculus and geometry.

Tangency to a Sphere

A line is tangent to a sphere if it touches the sphere at exactly one point and does not intersect it. For a sphere with center at point ( \mathbf{C} ) and radius ( r ), and a line in vector form given by

[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} ]

where ( \mathbf{a} ) is a point on the line, ( \mathbf{b} ) is the direction vector of the line, and ( \lambda ) is a scalar parameter, the condition of tangency is that the distance from the center of the sphere to the line is equal to the radius of the sphere.

Formula for Tangency to a Sphere

The condition of tangency to a sphere can be expressed as:

[ d = \frac{|\mathbf{b} \times (\mathbf{a} - \mathbf{C})|}{|\mathbf{b}|} = r ]

where ( d ) is the perpendicular distance from the center of the sphere to the line.

Tangency to a Plane

A line is tangent to a plane if it touches the plane at exactly one point and does not intersect it. For a plane with the equation

[ \mathbf{n} \cdot (\mathbf{r} - \mathbf{d}) = 0 ]

where ( \mathbf{n} ) is the normal vector to the plane, ( \mathbf{r} ) is any point on the plane, and ( \mathbf{d} ) is a specific point on the plane, the condition of tangency is that the line must be perpendicular to the normal vector of the plane.

Formula for Tangency to a Plane

The condition of tangency to a plane can be expressed as:

[ \mathbf{b} \cdot \mathbf{n} = 0 ]

where ( \mathbf{b} ) is the direction vector of the line.

Differences and Important Points

Here is a table summarizing the differences and important points regarding the condition of tangency for a sphere and a plane:

Aspect	Sphere	Plane
Surface Equation	( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 )	( Ax + By + Cz + D = 0 )
Vector Form	(	\mathbf{b} \times (\mathbf{a} - \mathbf{C})
Geometric Meaning	Line touches sphere at one point	Line touches plane at one point
Distance/Perpendicularity	Perpendicular distance from center to line equals radius	Line direction is perpendicular to plane normal

Examples

Example 1: Tangency to a Sphere

Given a sphere with center ( \mathbf{C} = (1, 2, 3) ) and radius ( r = 5 ), and a line with direction vector ( \mathbf{b} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k} ) passing through the point ( \mathbf{a} = (3, 0, 5) ), determine if the line is tangent to the sphere.

Solution:

First, we calculate ( \mathbf{a} - \mathbf{C} ):

[ \mathbf{a} - \mathbf{C} = (3 - 1)\mathbf{i} + (0 - 2)\mathbf{j} + (5 - 3)\mathbf{k} = 2\mathbf{i} - 2\mathbf{j} + 2\mathbf{k} ]

Next, we find the cross product ( \mathbf{b} \times (\mathbf{a} - \mathbf{C}) ):

[ \mathbf{b} \times (\mathbf{a} - \mathbf{C}) = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & 2 & 2 \ 2 & -2 & 2 \ \end{vmatrix} = 8\mathbf{i} - 2\mathbf{j} - 6\mathbf{k} ]

Now, we calculate the magnitude of the cross product and compare it to ( r|\mathbf{b}| ):

[ |\mathbf{b} \times (\mathbf{a} - \mathbf{C})| = \sqrt{8^2 + (-2)^2 + (-6)^2} = \sqrt{64 + 4 + 36} = \sqrt{104} ]

[ |\mathbf{b}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 ]

[ r|\mathbf{b}| = 5 \times 3 = 15 ]

Since ( \sqrt{104} \neq 15 ), the line is not tangent to the sphere.

Example 2: Tangency to a Plane

Given a plane with normal vector ( \mathbf{n} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} ) and a line with direction vector ( \mathbf{b} = 2\mathbf{i} + 6\mathbf{j} - 4\mathbf{k} ), determine if the line is tangent to the plane.

Solution:

We calculate the dot product ( \mathbf{b} \cdot \mathbf{n} ):

[ \mathbf{b} \cdot \mathbf{n} = (2)(3) + (6)(-1) + (-4)(2) = 6 - 6 - 8 = -8 ]

Since ( \mathbf{b} \cdot \mathbf{n} \neq 0 ), the line is not tangent to the plane.

These examples illustrate how to apply the condition of tangency in vector form to determine whether a line is tangent to a sphere or a plane. Understanding these conditions is essential for solving problems in 3D geometry and vector calculus.