Maths-
General
Easy
Question
The direction cosines of the line joining the points (4, 3, - 5) and (-2, 1, -8) are
- < 2, 4, -13 >
- < 6, 2, 3 >
- < 6/7, 2/7, 3/7 >
- None of these
Hint:
We are given the coordinates of two points. We have to find the direction cosines of the line joining the two points. Direction cosines are the values of cosines of the given line. They are the cosines of angles it makes with the three axis.
The correct answer is: < 6/7, 2/7, 3/7 >
We will denote the given point by P and Q.
P = (4,3,-5)
Q = (-2,1,-8)
Direction cosines are denoted by l, m, and n.
To find the direction cosines, we take the difference between the coordinates and the distance between two points. It is the ratio of direction ratio and distance between the points.
![fraction numerator x subscript 2 minus x subscript 1 over denominator P Q end fraction comma space fraction numerator y subscript 2 space minus y subscript 1 over denominator P Q end fraction comma fraction numerator z subscript 2 minus z subscript 1 over denominator P Q end fraction](data:image/png;base64,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)
We will find the distance between the given points using the distance formula.
![P Q space equals square root of left parenthesis x subscript 1 minus x subscript 2 right parenthesis squared plus left parenthesis y subscript 1 space minus space y subscript 2 right parenthesis squared plus left parenthesis z subscript 1 space minus z subscript 2 right parenthesis squared end root
space space space space space space space equals square root of left square bracket 4 minus left parenthesis negative 2 right parenthesis right square bracket squared plus left parenthesis 3 minus 1 right parenthesis to the power of 2 end exponent plus left square bracket left parenthesis negative 5 right parenthesis space minus left parenthesis negative 8 right parenthesis right square bracket squared end root
space space space space space space space equals square root of left parenthesis 4 space plus space 2 right parenthesis squared space plus space left parenthesis 3 space minus 1 right parenthesis squared plus space left parenthesis negative 5 plus 8 right parenthesis squared end root
space space space space space space space equals square root of 6 squared plus 2 squared plus 3 squared end root space
space space space space space space space equals square root of 36 plus 4 plus 9 end root
space space space space space space space equals square root of 49
space space space space space space space space equals 7
P Q space equals space 7](data:image/png;base64,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Direction ratios are as follows:
x1 - x2 = 4 - (-2)
= 4 + 2
= 6
y1 - y2 = 3 - 2
= 1
z1 - z2 = -5 - (-8)
= -5 + 8
= 3
ubstituting the values,
The direction cosines of the line joining two points are as follows:
![l comma space m comma space n space equals space fraction numerator x subscript 1 minus x subscript 2 over denominator P Q end fraction comma fraction numerator y subscript 1 minus y subscript 2 over denominator P Q end fraction comma fraction numerator z subscript 1 minus z subscript 2 over denominator P Q end fraction
space space space space space space space space space space space space equals 6 over 7 comma 2 over 7 comma 3 over 7
S o comma space t h e space r i g h t space o p t i o n space i s space less than 6 over 7 comma 2 over 7 comma 3 over 7 greater than](data:image/png;base64,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)
For such questions, we should know formula to find direction cosines.
Related Questions to study
Maths-
The locus of x2 + y2 + z2 = 0 is
For such questions, we have to be careful about points satisfying the equation.
The locus of x2 + y2 + z2 = 0 is
Maths-General
For such questions, we have to be careful about points satisfying the equation.
Maths-
Maths-General
Maths-
Maths-General
Maths-
Let f (x) be a quadratic expression which is positive for all real x. If g(x) = f (x) + f ¢ (x) + f ¢¢ (x), then for any real x
Let f (x) be a quadratic expression which is positive for all real x. If g(x) = f (x) + f ¢ (x) + f ¢¢ (x), then for any real x
Maths-General
Maths-
Maths-General
Maths-
=
=
Maths-General
Maths-
Maths-General
Maths-
Maths-General
Maths-
For such questions, we should know different the condition of limit to exist.
Maths-General
For such questions, we should know different the condition of limit to exist.
Maths-
then sinx =
then sinx =
Maths-General
Maths-
If
then x=
If
then x=
Maths-General
Maths-
If 2
= x then x =
If 2
= x then x =
Maths-General
Maths-
I :
+ ![T a n to the power of negative 1 end exponent 3 equals fraction numerator text 3π end text over denominator text 4 end text end fraction](data:image/png;base64,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)
II :
= 0
Which of the above statements is correct?
I :
+ ![T a n to the power of negative 1 end exponent 3 equals fraction numerator text 3π end text over denominator text 4 end text end fraction](data:image/png;base64,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)
II :
= 0
Which of the above statements is correct?
Maths-General
Chemistry-
The solubility product of a salt having general formula
in water is
. The concentration of
ions in the aqueous solution of the salt is:
The solubility product of a salt having general formula
in water is
. The concentration of
ions in the aqueous solution of the salt is:
Chemistry-General
Chemistry-
the-dissociation of weak electrolyte (weak acid) is expressed in terms of Ostwald dilution law stronger is the acid, weaker is its conjugate base_ The dissociation constants of an acid (
) and its conjugate base (
) are related by the given relation: -
![K subscript w end subscript equals K subscript a end subscript cross times K subscript b end subscript](data:image/png;base64,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)
At
(Ionic product of water) ![equals 10 to the power of negative 14 end exponent](data:image/png;base64,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)
Phosphoric acid isa weak acid. It is used in fertilizer, food, detergent and toothpaste. Structure of phosphoric acid is:
![](data:image/png;base64,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)
Aqueous solution of phosphoric acid with a density oflg mL-1 containing 0.05% by weight of phosphoric acid is used to impart tart taste to many soft drinks. Phosphate ion is an inteife ring radical in qualitative analysis. It should be removed for analysis beyond third group of qualitative analysis
the basicity of phosphoric acid is:
the-dissociation of weak electrolyte (weak acid) is expressed in terms of Ostwald dilution law stronger is the acid, weaker is its conjugate base_ The dissociation constants of an acid (
) and its conjugate base (
) are related by the given relation: -
![K subscript w end subscript equals K subscript a end subscript cross times K subscript b end subscript](data:image/png;base64,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)
At
(Ionic product of water) ![equals 10 to the power of negative 14 end exponent](data:image/png;base64,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)
Phosphoric acid isa weak acid. It is used in fertilizer, food, detergent and toothpaste. Structure of phosphoric acid is:
![](data:image/png;base64,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)
Aqueous solution of phosphoric acid with a density oflg mL-1 containing 0.05% by weight of phosphoric acid is used to impart tart taste to many soft drinks. Phosphate ion is an inteife ring radical in qualitative analysis. It should be removed for analysis beyond third group of qualitative analysis
the basicity of phosphoric acid is:
Chemistry-General