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Angle between 2 Lines - Three Dimensional Geometry, Class 12, Math Video Lecture | Mathematics (Maths) Class 12 - JEE

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FAQs on Angle between 2 Lines - Three Dimensional Geometry, Class 12, Math Video Lecture - Mathematics (Maths) Class 12 - JEE

1. What is the formula to calculate the angle between two lines in three-dimensional geometry?
Ans. The formula to calculate the angle between two lines in three-dimensional geometry is given by the dot product of the direction vectors of the two lines divided by the product of their magnitudes. Mathematically, it can be expressed as: cosθ = (a1 * b1 + a2 * b2 + a3 * b3) / (sqrt(a1^2 + a2^2 + a3^2) * sqrt(b1^2 + b2^2 + b3^2)) Here, (a1, a2, a3) and (b1, b2, b3) are the direction vectors of the two lines.
2. How do you find the direction vectors of two lines in three-dimensional geometry?
Ans. To find the direction vectors of two lines, we can choose any two points on each line and subtract the coordinates of the first point from the coordinates of the second point. The resulting vector will be the direction vector of the line. For example, if we have two points A(x1, y1, z1) and B(x2, y2, z2) on a line, the direction vector of the line can be calculated as: (a1, a2, a3) = (x2 - x1, y2 - y1, z2 - z1)
3. Can the angle between two lines be greater than 90 degrees?
Ans. No, the angle between two lines in three-dimensional geometry cannot be greater than 90 degrees. The angle between two lines is always measured within the range of 0 to 90 degrees. If the dot product of the direction vectors of the two lines is positive, the angle between them will be acute (less than 90 degrees). If the dot product is negative, the angle will be obtuse (greater than 90 degrees). If the dot product is zero, the lines are perpendicular to each other and the angle between them is 90 degrees.
4. What does it mean if the angle between two lines is zero degrees?
Ans. If the angle between two lines is zero degrees, it means that the two lines are parallel to each other. In three-dimensional geometry, parallel lines never intersect, and their direction vectors are scalar multiples of each other. This means that the dot product of their direction vectors will be equal to the product of their magnitudes. Therefore, when the angle between two lines is zero degrees, the dot product of their direction vectors will be equal to the product of their magnitudes, resulting in cosθ = 1.
5. How can the angle between two lines be used in real-world applications?
Ans. The concept of the angle between two lines in three-dimensional geometry has various real-world applications. For example: - In computer graphics, the angle between two lines can be used to determine the orientation of objects or to calculate the lighting and shading effects on a 3D model. - In physics, the angle between two lines can be used to analyze the motion of objects in three-dimensional space, such as the trajectory of a projectile or the movement of celestial bodies. - In engineering, the angle between two lines can be used to determine the alignment of structures, such as beams or columns, and to calculate the forces acting on them. - In navigation, the angle between two lines can be used to determine the direction and orientation of an object or a vehicle relative to a reference point or a target. - In robotics, the angle between two lines can be used to calculate the movement and positioning of robot arms or manipulators, enabling precise control and coordination.
204 videos|288 docs|139 tests

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