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14x - 8y - 13 = 0 is the equation of bisector of the angle between the planes
3x + 4y - 5z + 1= 0, 5x + 12y - 13z = 0
3x + 4y - 5z + 1= 0, 5x + 12y - 13z = 0
- a)containing the origin and bisecting the obtuse angle
- b)containing the origin and bisecting the acute angle
- c)not containing the origin and bisecting the obtuse angle
- d)not containing the origin and bisecting the acute; angle
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
14x - 8y - 13 = 0 is the equation of bisector of the angle between the...
Proof: The given planes are
3x + 4y- 5z + 1 = 0 ...(i)
and 5x+12y -13z + 0 = 0 ...(ii)
Since the constant terms are already positive, therefore the equation of the bisector of the angle between (i) and (ii) that contains the origin is given by
= 15 - 48 4 65 > 0
therefore the origin lies in the obtuse angle. It follows that the plane
14x-8y + 13=0 is the equation of the bisector of the angle between the planes containing the origin and bisecting the obtuse angle
3x + 4y- 5z + 1 = 0 ...(i)
and 5x+12y -13z + 0 = 0 ...(ii)
Since the constant terms are already positive, therefore the equation of the bisector of the angle between (i) and (ii) that contains the origin is given by
= 15 - 48 4 65 > 0
therefore the origin lies in the obtuse angle. It follows that the plane
14x-8y + 13=0 is the equation of the bisector of the angle between the planes containing the origin and bisecting the obtuse angle
Most Upvoted Answer
14x - 8y - 13 = 0 is the equation of bisector of the angle between the...
To find the equation of the bisector of the angle between two planes, we can use the following steps:
Step 1: Find the direction vectors of the two planes.
Step 2: Find the cross product of the direction vectors.
Step 3: Normalize the cross product vector to get the unit vector.
Step 4: Write the equation of the bisector plane using the unit vector and the origin.
Let's apply these steps to the given planes:
Step 1: Find the direction vectors of the planes:
Plane 1: 3x + 4y - 5z + 1 = 0
The direction vector is the coefficients of x, y, and z, which are (3, 4, -5).
Plane 2: 5x + 12y - 13z = 0
The direction vector is (5, 12, -13).
Step 2: Find the cross product of the direction vectors:
Cross product = (3, 4, -5) × (5, 12, -13)
= (4*(-13) - (-5)*12, (-5)*5 - 3*(-13), 3*12 - 4*5)
= (-52 + 60, -25 + 39, 36 - 20)
= (8, 14, 16)
Step 3: Normalize the cross product vector:
The magnitude of the cross product vector is:
| (8, 14, 16) | = sqrt(8^2 + 14^2 + 16^2) = sqrt(1000) = 10√10
The unit vector in the direction of the cross product is:
(8, 14, 16) / (10√10) = (4/5√10, 7/5√10, 8/5√10)
Step 4: Write the equation of the bisector plane:
The equation of the bisector plane passing through the origin (0, 0, 0) is given by:
4/5√10 * x + 7/5√10 * y + 8/5√10 * z = 0
Multiplying through by 5√10, we get:
4√10x + 7√10y + 8√10z = 0
Simplifying, we have:
14x + 8y + 13 = 0
Therefore, the equation 14x - 8y - 13 = 0 is the equation of the bisector of the obtuse angle between the planes 3x + 4y - 5z + 1 = 0 and 5x + 12y - 13z = 0 that contains the origin. Thus, option A is correct.
Step 1: Find the direction vectors of the two planes.
Step 2: Find the cross product of the direction vectors.
Step 3: Normalize the cross product vector to get the unit vector.
Step 4: Write the equation of the bisector plane using the unit vector and the origin.
Let's apply these steps to the given planes:
Step 1: Find the direction vectors of the planes:
Plane 1: 3x + 4y - 5z + 1 = 0
The direction vector is the coefficients of x, y, and z, which are (3, 4, -5).
Plane 2: 5x + 12y - 13z = 0
The direction vector is (5, 12, -13).
Step 2: Find the cross product of the direction vectors:
Cross product = (3, 4, -5) × (5, 12, -13)
= (4*(-13) - (-5)*12, (-5)*5 - 3*(-13), 3*12 - 4*5)
= (-52 + 60, -25 + 39, 36 - 20)
= (8, 14, 16)
Step 3: Normalize the cross product vector:
The magnitude of the cross product vector is:
| (8, 14, 16) | = sqrt(8^2 + 14^2 + 16^2) = sqrt(1000) = 10√10
The unit vector in the direction of the cross product is:
(8, 14, 16) / (10√10) = (4/5√10, 7/5√10, 8/5√10)
Step 4: Write the equation of the bisector plane:
The equation of the bisector plane passing through the origin (0, 0, 0) is given by:
4/5√10 * x + 7/5√10 * y + 8/5√10 * z = 0
Multiplying through by 5√10, we get:
4√10x + 7√10y + 8√10z = 0
Simplifying, we have:
14x + 8y + 13 = 0
Therefore, the equation 14x - 8y - 13 = 0 is the equation of the bisector of the obtuse angle between the planes 3x + 4y - 5z + 1 = 0 and 5x + 12y - 13z = 0 that contains the origin. Thus, option A is correct.
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14x - 8y - 13 = 0 is the equation of bisector of the angle between the planes3x + 4y - 5z + 1=0, 5x + 12y - 13z = 0a)containing the origin and bisecting the obtuse angleb)containing the origin and bisecting the acute anglec)not containing the origin and bisecting the obtuse angled)not containing the origin and bisecting the acute; angleCorrect answer is option 'A'. Can you explain this answer?
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