Angle between a line and a plane

In this article, we will discuss the concept of the angle between a line and a plane. We will discuss how to find the angle between the line and the plane and the angle between the line and plane formula. Along with this concept, several other concepts like relations between cosine of a line, equation of a line in space, equation of a line passing through two given points, and angle between two lines are discussed in this chapter. A straight line refers to a two-dimensional figure with an infinite length with no width or height. The straight line, part of 2 D geometry, has an infinite number of points extending in any direction.

How do you find the angle between the line and the plane?

Definition of a plane- A plane is a two-dimensional figure with length and width but no height. When an infinite number of points extend in any direction, the resultant 2-D figure refers to a flat surface called a plane.

Example of straight line and a plane- When a kid draws a connecting line point A from point B, then that line is a straight line and the surface on which he drew that a straight line is a plane.

Definition of an angle- When two lines or surfaces intersect each other in space, they form an angle. When a normal is drawn from where the line touches it on a plane, an angle between a line and a plane is formed. The complement of the angle formed between a normal and a line is equal to the angle between a line and a plane.

The angle formed between the line and the plane will be different in each of the three scenarios in which a line and a plane can exist together. For example, the angle between a line and a plane will be 0 degrees if the straight line is drawn on the plane or parallel.

If the straight line is secant to the plane, then the angle formed between the line and the plane is the angle formed by the line with its orthogonal projection on the plane. (Orthogonal projection refers to the projection method in which parallel lines are used to depict the outline of an object on a plane.) The symbol ‘ α represents the angle formed by the line which is secant to the plane.’

The angle between the line and plane formula

The first formula to find the angle between the line and the plane is – The Cartesian Form.

The equation of a straight line in cartesian form is –

(x – x₁)/ a = (y – y₁)/ b = (z – z₁)/ c, where, x₁, y₁, and z₁ represent the coordinates of a point in the straight line.

The equation of a plane in cartesian form is expressed as –

a₂x + b₂y + c₂z + d₂ = 0, where, x₂, y₂, and z₂ represent the coordinates of a point in the plane.

The formula for the angle between the line and the plane is given by –

Sin ɵ = (a₁a₂ + b₁b₂ + c₁c₂)/  a₁² + b₁² + c₁² ). ( a₂² + b₂² + c₂²).

The second formula to find the angle between the line and the plane is – The Vector Form.

The equation of a plane in vector form is expressed as, r = a λ +b.

The equation of a line in vector form is expressed as, r *.n = d.

The angle between the line and the plane in vector form is given by –

Sin ɵ = n * b / |n| |b|.

The angle between vector and plane solved examples

Example 1 – Find the angle between the straight line (x + 1) / 2 = y/ 3 = (z – 3)/ 6 and the plane 10x + 2y – 11z = 3.

 It is given that,

(x + 1) / 2 = y / 3 = (z – 3) / 6, and 10x + 2y – 11z = 3.

The direction ratios of the line = (2, 3, 6)

And, the direction ratios of the normal to the plane = (10, 2, -11).

The angle between the line and the plane can be expressed as –

Sin-1 [{(a x a’) + (b x b’) + (c x c’)} / {Sq. root (a² + b² + c²) * sq. root (a’² + b’² + c’²)}]

Here, a, b , c are direction ratios of the line and are equal to 2, 3, 6 respectively.

And, a’, b’, c, are direction ratios of the normal to the plane and are equal to 10 2, -11 respectively.

On substituting the values, we get

The angle between the line and the plane = sin-1 [{2 x 10 + 3 x 2 + 6 x (-11)} / {sq. root (2² + 3² + 6²) * sq. root (10² + 2² + (-11)²)}]

= sin^-1 [(20 + 6 – 66) / 7 * 15]

= sin^-1 [(-40) / 105] 

= sin^-1 (- 8 / 21).

On taking only the magnitude,

Angle = sin^-1 (8 / 21).

Conclusion

Through this concise and precisely written article, the students have been provided with a study guide covering topics such as – angle between the line and the plane, the angle between vector and plane solved examples, and the angle between the line and the plane formula. The article is well-written in an easy-to-understand manner to help students grasp the concepts easily and prepare them for CBSE exams. Various examples, formulas, and frequently asked questions are discussed for the very purpose of aiding students’ preparation.