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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.
Text Transcript from Video
angle between two lines lit l1 and l2 be
two lines passing through origin with
Direction ratios a 1 B 1 C 1 and a 2 B 2
C 2 respectively let P be any point on
l1 and Q be any point on l2 let theta be
the angle between two directed lines or
P and Q now these directed line segments
are nothing but the vectors with the
components a 1 B 1 C 1 and a 2 B 2 C 2
respectively therefore the angle between
the vectors o P and O Q R cos theta is
equal to modulus of a 1 a 2 plus B 1 B 2
plus even C 2 by root over a 1 square
plus B 1 square plus C 1 square into
root over a 2 square plus B 2 square
plus C 2 square if instead of Direction
ratios Direction cosines say L 1 M 1 n 1
for l1 and l2 M 2 n 2 for l2 all given
then the angle between these two lines
are given as cos theta is equal to
modulus of L 1 M 2 plus M 1 M 2 plus N 1
n 2 past L 1 square plus M 1 square plus
N 1 squared and L 2 square plus M 2
square plus n 2 squared both are equal
to 1 angle between two straight lines in
terms of sine theta we know sine theta
is equal to root over 1 minus cos square
theta substituting the value of cos
theta we get sine theta
is equal to root over 1 minus a 1 a 2
plus B 1 B 2 plus C 1 C 2 whole square
but D 1 square plus B 1 square plus C 1
square into a 2 square plus B 2 square
plus C 2 square on simplification we get
sine theta is equal to a 1 B 2 minus a 2
B 1 whole square plus B 1 C 2 minus B 2
C 1 whole square plus C 1 a 2 minus C 2
a 1 whole square divided by root over K
1 square plus B 1 square plus C 1 square
into root over a 2 square plus B 2
square plus C 2 square condition firm
two lands with the direction ratios a 1
B 1 C 1 and a 2 B 2 C 2 to be
perpendicular if these two lines are
perpendicular then the angle between
them that is theta is equal to 90
degrees then cos 90 is equal to 0 so we
get a 1 a 2 plus B 1 B 2 plus C 1 C 2 is
equal to 0 condition for the two lines
to be parallel if two lines are parallel
then angle between them is 0
substituting theta is equal to 0 in the
formula sine theta we get the condition
for two lines to be parallel as a1 / a2
is equal to b1 / b2 is equal to c1 over
c2 now let us fire
the angle between two lines when their
equations are given if theta is acute
angle between the lines our vector is
equal to K one vector plus lambda B 1
vector and our vector is equal to theta
vector plus mu into B 2 vector then the
angle between these two straight lines
this cos theta is equal to modulus of B
1 dot v2 over not B 1 into mod B 2
two lines passing through origin with
Direction ratios a 1 B 1 C 1 and a 2 B 2
C 2 respectively let P be any point on
l1 and Q be any point on l2 let theta be
the angle between two directed lines or
P and Q now these directed line segments
are nothing but the vectors with the
components a 1 B 1 C 1 and a 2 B 2 C 2
respectively therefore the angle between
the vectors o P and O Q R cos theta is
equal to modulus of a 1 a 2 plus B 1 B 2
plus even C 2 by root over a 1 square
plus B 1 square plus C 1 square into
root over a 2 square plus B 2 square
plus C 2 square if instead of Direction
ratios Direction cosines say L 1 M 1 n 1
for l1 and l2 M 2 n 2 for l2 all given
then the angle between these two lines
are given as cos theta is equal to
modulus of L 1 M 2 plus M 1 M 2 plus N 1
n 2 past L 1 square plus M 1 square plus
N 1 squared and L 2 square plus M 2
square plus n 2 squared both are equal
to 1 angle between two straight lines in
terms of sine theta we know sine theta
is equal to root over 1 minus cos square
theta substituting the value of cos
theta we get sine theta
is equal to root over 1 minus a 1 a 2
plus B 1 B 2 plus C 1 C 2 whole square
but D 1 square plus B 1 square plus C 1
square into a 2 square plus B 2 square
plus C 2 square on simplification we get
sine theta is equal to a 1 B 2 minus a 2
B 1 whole square plus B 1 C 2 minus B 2
C 1 whole square plus C 1 a 2 minus C 2
a 1 whole square divided by root over K
1 square plus B 1 square plus C 1 square
into root over a 2 square plus B 2
square plus C 2 square condition firm
two lands with the direction ratios a 1
B 1 C 1 and a 2 B 2 C 2 to be
perpendicular if these two lines are
perpendicular then the angle between
them that is theta is equal to 90
degrees then cos 90 is equal to 0 so we
get a 1 a 2 plus B 1 B 2 plus C 1 C 2 is
equal to 0 condition for the two lines
to be parallel if two lines are parallel
then angle between them is 0
substituting theta is equal to 0 in the
formula sine theta we get the condition
for two lines to be parallel as a1 / a2
is equal to b1 / b2 is equal to c1 over
c2 now let us fire
the angle between two lines when their
equations are given if theta is acute
angle between the lines our vector is
equal to K one vector plus lambda B 1
vector and our vector is equal to theta
vector plus mu into B 2 vector then the
angle between these two straight lines
this cos theta is equal to modulus of B
1 dot v2 over not B 1 into mod B 2
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Oct 31, 2024 Last updated |
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