Find the distance of a point (2, 5, −3) from the plane  from

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 Multiple Choice QuestionsLong Answer Type

301. Show that the lines:
fraction numerator straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y minus 7 over denominator 4 end fraction space equals fraction numerator straight z plus 3 over denominator negative 5 end fraction comma space space fraction numerator straight x minus 8 over denominator 7 end fraction space equals space fraction numerator straight y minus 4 over denominator 1 end fraction space equals space fraction numerator straight z minus 5 over denominator 3 end fraction are co-planar, find their common point and the equation of the plane in which way they lie. 
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 Multiple Choice QuestionsShort Answer Type

302. Find the angle between each of the following pairs of lines:
A line with direction ratios 2 : 2 : 1.
A line joining (3, 1, 4) to (7, 2, 12)
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303. Find the angle between each of the following pairs of lines:
5, –12, 13;    – 3, 4, 5
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304. Find the angle between each of the following pairs of lines:
1, 1, 2;   square root of 3 minus 1 comma space space minus square root of 3 minus 1 comma space 4
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305.

Write the sum of intercepts cut off by the planestraight r with rightwards arrow on top. open parentheses 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top close parentheses minus 5 space equals 0 on the three axes.

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306.

Find the coordinates of the foot of perpendicular drawn from the point A
(-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.

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 Multiple Choice QuestionsLong Answer Type

307.

Find the equation of the plane which contains the line of intersection of the planes
straight r with rightwards arrow on top. left parenthesis straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top right parenthesis minus 4 space equals space 0
and
straight r with rightwards arrow on top. left parenthesis negative 2 straight i with hat on top plus straight j with hat on top plus straight k with hat on top right parenthesis plus 5 space space equals space 0

and whose intercept on the x-axis is equal to that of on y-axis.

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 Multiple Choice QuestionsShort Answer Type

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308.

Find the distance of a point (2, 5, −3) from the plane straight r with rightwards arrow on top. open parentheses 6 straight i with hat on top minus 3 straight j with hat on top plus 2 straight k with hat on top close parentheses space equals 4.


Consider the vector equation of the plane.
straight r with rightwards arrow on top. space open parentheses 6 straight i with hat on top minus 3 straight j with hat on top space plus space 2 straight k with hat on top close parentheses space equals space 4
space space space space space rightwards double arrow open parentheses straight x straight i with hat on top space plus space straight y straight j with hat on top space plus space straight z straight k with hat on top close parentheses. space open parentheses 6 straight i with hat on top minus 3 straight j with hat on top space plus space 2 straight k with hat on top close parentheses space equals space 4
space space space space space rightwards double arrow 6 straight x minus 37 plus 2 straight z space equals 4
space space space space space rightwards double arrow 6 straight x minus 3 straight y plus 2 straight z minus 4 space equals space 0 space space space space space
space space
Thus the Cartesian equation of the plane is
6x-3y+2z-4 = 0
Let d be the distance between the point (2, 5, -3)
Thus space straight d space equals space open vertical bar fraction numerator ax subscript 1 plus by subscript 1 plus cz subscript 1 plus straight d over denominator square root of straight a squared plus straight b squared plus straight c squared end root end fraction close vertical bar
space space space rightwards double arrow straight d space equals space open vertical bar fraction numerator 6 cross times 2 minus 3 cross times 5 plus 2 cross times left parenthesis negative 3 right parenthesis minus 4 over denominator square root of 6 squared plus left parenthesis negative 3 right parenthesis squared plus 2 squared end root end fraction close vertical bar
space space space rightwards double arrow straight d space equals space open vertical bar fraction numerator 12 minus 15 minus 6 minus 4 over denominator square root of 36 plus 9 plus 4 end root end fraction close vertical bar
space space space space rightwards double arrow straight d space equals space open vertical bar fraction numerator negative 13 over denominator square root of 49 end fraction close vertical bar
space space space space rightwards double arrow straight d space equals space 13 over 7 units

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309.

Find x such that the four points A(4, 1, 2), B(5, x, 6) , C(5, 1, -1) and D(7, 4, 0) are coplanar.

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310.

A line passing through the point A with position vector straight a with rightwards arrow on top space equals space 4 straight i with hat on top plus 2 straight j with hat on top plus 2 straight k with hat on top is parallel to the vector straight b with rightwards arrow on top space equals space 2 straight i with hat on top plus 3 straight j with hat on top plus 6 straight k with hat on top. Find the length of the perpendicular drawn on this line from a point p with vector stack straight r subscript 1 with rightwards arrow on top space equals straight i with hat on top plus 2 straight j with hat on top plus 3 straight k with hat on top.

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