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Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?
- a)(3, 4. -2)
- b)(4, -2, 5)
- c)(4, 5, -2)
- d)(2, 4, 5)
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) an...
Equation of BC is
⇒ x = 2λ , y = 8λ − 11, z = −3λ + 4
Now, 2(x − 1) + 8(y − 8) − 3(z − 4) = 0
⇒ 2x + 8y − 3z = 54
⇒ 4λ + 64λ − 88 + 9λ − 12 = 54
⇒ λ = 2
∴ foot = (4, 5, −2)
⇒ x = 2λ , y = 8λ − 11, z = −3λ + 4
Now, 2(x − 1) + 8(y − 8) − 3(z − 4) = 0
⇒ 2x + 8y − 3z = 54
⇒ 4λ + 64λ − 88 + 9λ − 12 = 54
⇒ λ = 2
∴ foot = (4, 5, −2)
Most Upvoted Answer
Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) an...
Solution:
To find the coordinates of point D, we need to first find the equation of line BC, then find the point of intersection of line AD and line BC, which will be the point D.
Finding equation of line BC:
Let's find the direction ratios of line BC:
x-coordinate of B = 0, x-coordinate of C = 2
Direction ratio of BC along x-axis = 2 - 0 = 2
y-coordinate of B = -11, y-coordinate of C = -3
Direction ratio of BC along y-axis = -3 - (-11) = 8
z-coordinate of B = 4, z-coordinate of C = 1
Direction ratio of BC along z-axis = 1 - 4 = -3
So the direction ratios of line BC are (2, 8, -3)
Now we can find the equation of line BC passing through point B as:
(x - 0)/2 = (y + 11)/8 = (z - 4)/(-3)
Finding point of intersection of line AD and line BC:
Let's first find the direction ratios of line AD:
Direction ratios of AD will be same as the direction ratios of line perpendicular to plane BAC, passing through A.
The normal to plane BAC can be found by taking cross product of vectors AB and AC:
AB = (0-1)i + (-11-8)j + (4-4)k = -i - 19j
AC = (2-1)i + (-3-8)j + (1-4)k = i - 11j - 3k
Normal to plane BAC = AB x AC = 52i - 4j - 18k (after simplification)
So the direction ratios of line AD will be (52, -4, -18)
Now we can find the equation of line AD passing through point A as:
(x - 1)/52 = (y - 8)/(-4) = (z - 4)/(-18)
To find the point of intersection of line AD and line BC, we need to solve these two equations. One way to do that is to equate the two ratios of (x - 1)/52 and (y - 8)/(-4) from the two equations and solve for x and y. Then use one of the two ratios to find z.
(x - 1)/52 = (y - 8)/(-4)
=> x = (-13/26)y + (385/26)
Substituting this in the equation of line BC:
((-13/26)y + (385/26) - 0)/2 = (y + 11)/8
=> y = -26/5
Using the ratio (z - 4)/(-18) from the equation of line AD:
(z - 4)/(-18) = (y - 8)/(-4)
=> z = 4 - (9/5)(-26/5) = -2/5
So the coordinates of point D are (4, 5, -2), which matches with option C.
To find the coordinates of point D, we need to first find the equation of line BC, then find the point of intersection of line AD and line BC, which will be the point D.
Finding equation of line BC:
Let's find the direction ratios of line BC:
x-coordinate of B = 0, x-coordinate of C = 2
Direction ratio of BC along x-axis = 2 - 0 = 2
y-coordinate of B = -11, y-coordinate of C = -3
Direction ratio of BC along y-axis = -3 - (-11) = 8
z-coordinate of B = 4, z-coordinate of C = 1
Direction ratio of BC along z-axis = 1 - 4 = -3
So the direction ratios of line BC are (2, 8, -3)
Now we can find the equation of line BC passing through point B as:
(x - 0)/2 = (y + 11)/8 = (z - 4)/(-3)
Finding point of intersection of line AD and line BC:
Let's first find the direction ratios of line AD:
Direction ratios of AD will be same as the direction ratios of line perpendicular to plane BAC, passing through A.
The normal to plane BAC can be found by taking cross product of vectors AB and AC:
AB = (0-1)i + (-11-8)j + (4-4)k = -i - 19j
AC = (2-1)i + (-3-8)j + (1-4)k = i - 11j - 3k
Normal to plane BAC = AB x AC = 52i - 4j - 18k (after simplification)
So the direction ratios of line AD will be (52, -4, -18)
Now we can find the equation of line AD passing through point A as:
(x - 1)/52 = (y - 8)/(-4) = (z - 4)/(-18)
To find the point of intersection of line AD and line BC, we need to solve these two equations. One way to do that is to equate the two ratios of (x - 1)/52 and (y - 8)/(-4) from the two equations and solve for x and y. Then use one of the two ratios to find z.
(x - 1)/52 = (y - 8)/(-4)
=> x = (-13/26)y + (385/26)
Substituting this in the equation of line BC:
((-13/26)y + (385/26) - 0)/2 = (y + 11)/8
=> y = -26/5
Using the ratio (z - 4)/(-18) from the equation of line AD:
(z - 4)/(-18) = (y - 8)/(-4)
=> z = 4 - (9/5)(-26/5) = -2/5
So the coordinates of point D are (4, 5, -2), which matches with option C.
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Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer?
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Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer?.
Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer?.
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Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer?, a detailed solution for Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of Let the coordinates of the points A, B, C be (1, 8, 4), (0, -11, 4) and (2, -3, 1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?a)(3, 4. -2)b)(4, -2, 5)c)(4, 5, -2)d)(2, 4, 5)Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
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