Introduction to Three Dimensional Geometry
3D geometry involves the mathematics of shapes in 3D space and involving 3 coordinates which are x-coordinate, y-coordinate and z-coordinate. In a 3d space, three parameters are required to find the exact location of a point. For JEE, three-dimensional geometry plays a major role as a lot of questions are included in the exam. Here, the basic concepts of geometry involving 3-dimensional coordinates are covered which will help to understand different operations on a point in 3d plane.
Coordinate System in 3D Geometry
In 3 dimensional geometry, a coordinate system refers to the process of identifying the position or location of a point in the coordinate plane. To understand more about coordinate planes and system, refer to the coordinate geometry lesson which covers all the basic concepts, theorems, and formulas related to coordinate or analytic geometry.
Rectangular coordinate system
Three lines perpendicular to each other pass through a common point. That common point is called the origin, the 3 lines the axes. They are x-axis, y-axis, z-axis respectively. O is the observer with respect to his position of any other point is measured. The position or coordinates of any point in 3D space is measured by how much he has moved along x, y and z-axis respectively. So if a point has a position (3, -4, 5) means he has moved 3 unit along positive x-axis, 4 unit along negative y-axis, 5 unit along positive z-axis.
Rectangular coordinate system – 3D Geometry
Distance from the Origin
Distance from the Origin in 3D Space – 3D Geometry
Distance from the origin. By using Pythagoras theorem. The distance of P(x, y, z) from the origin (0, 0, 0) is:
\(\begin{array}{l}\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}\end{array} \)
Distance between 2 points
Distance between 2 points P(x1, y1, z1) and Q(x2, y2, z2) is:
\(\begin{array}{l}PQ = \sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}+{{\left( {{z}_{2}}-{{z}_{1}} \right)}^{2}}}\end{array} \)
Division of a line joining 2 points
Let P(x1, y1, z1) and Q(x2, y2, z2) be 2 points. R derives the line segment PQ in ratio internally. Then R has coordinates;
\(\begin{array}{l}\left( \frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n}\frac{m{{z}_{2}}+n{{z}_{1}}}{m+n} \right)\end{array} \)
Projection in 3D Space
Projection in 3D Space – 3D Geometry
Let AB be a line segment. It’s projection on a line PQ, AB cos θ, where θ is the angle between AB and PQ or CD.
Direction cosines and direction Ratios of a Line in Cartesian Plane
Cosines of the angles a line makes with the positive x, y and z axis, respectively, are called direction cosines of that line.
Learn More: Direction Cosines & Direction Ratios Of A Line
So if those angles are α, β and γ, then cos α, cos β, and cos γ are the direction cosines of the line. They are denoted by l, m, n.
\(\begin{array}{l}{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1.\end{array} \)
(Proof will be given)
Any 3 numbers, a, b, and c, which are proportional to direction cosines, are called direction ratios.
Hence,
\(\begin{array}{l}\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=\frac{\sqrt{{{l}^{2}}+{{m}^{2}}+{{n}^{2}}}}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}=\frac{1}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}}\end{array} \)
\(\begin{array}{l}\therefore l=\frac{a}{\sqrt{\sum{{{a}^{2}}}}},m=\frac{b}{\sqrt{\sum{{{a}^{2}}}}},n=\frac{c}{\sqrt{\sum{{{a}^{2}}}}}\end{array} \)
Direction cosine of line joining two given points
Let P(x1, y1, z1) and Q(x2, y2, z2) be 2 points. Then, direction cosines will be
\(\begin{array}{l}l=\frac{{{x}_{2}}-{{x}_{1}}}{\left| PQ \right|},m=\frac{{{y}_{2}}-{{y}_{1}}}{\left| PQ \right|},n=\frac{{{z}_{2}}-{{z}_{1}}}{\left| PQ \right|}\end{array} \)
Projection of line segment joining 2 points, on another line
Consider P(x1, y1, z1) and Q(x2, y2, z2).
Projection of PQ on a line whose direction cosines are l, m, n is
\(\begin{array}{l}l\left( {{x}_{2}}-{{x}_{1}} \right)+m\left( {{y}_{2}}-{{y}_{1}} \right)+n\left( {{z}_{2}}-{{z}_{1}} \right)\end{array} \)
Angle between 2 lines in 3 Dimensional Space
2 lines having direction cosines (l1, m1, n1) and (l2, m2, n2). Then angle between them is
\(\begin{array}{l}\theta ={{\cos }^{-1}}\left( {{l}_{1}}{{l}_{2}}+{{m}_{1}}{{m}_{2}}+{{n}_{1}}{{n}_{2}} \right)\end{array} \)
Projection of a plane area on 3 coordinate planes
\(\begin{array}{l}\text{Let}\ \bar{A}\ \text{be the vector area.}\end{array} \)
If its direction cosines are cos α, cos β, and cos γ. Then projections are \(\begin{array}{l}{{A}_{1}}=A\cos \alpha ,{{A}_{2}}=A\cos \beta ,{{A}_{3}}=A\cos \gamma .\end{array} \)
\(\begin{array}{l}\therefore {{A}^{2}}=A_{1}^{2}+A_{2}^{2}+A_{3}^{2}\end{array} \)
Area of a triangle:
Using the projection formula, the area of a triangle
\(\begin{array}{l}=\frac{1}{4}{{\left| \begin{matrix} {{x}_{1}} & {{y}_{1}} & 1 \\ {{x}_{2}} & {{y}_{2}} & 1 \\ {{x}_{3}} & {{y}_{3}} & 1 \\ \end{matrix} \right|}^{2}}+\,\,\frac{1}{4}{{\left| \begin{matrix} {{y}_{1}} & {{z}_{1}} & 1 \\ {{y}_{2}} & {{z}_{2}} & 1 \\ {{y}_{3}} & {{z}_{3}} & 1 \\ \end{matrix} \right|}^{2}}+\,\,\frac{1}{4}{{\left| \begin{matrix} {{x}_{1}} & {{z}_{1}} & 1 \\ {{x}_{2}} & {{z}_{2}} & 1 \\ {{x}_{3}} & {{z}_{3}} & 1 \\ \end{matrix} \right|}^{2}}\end{array} \)
Check out more details about the area of a triangle in coordinate geometry , its derivation and problem solving strategies, etc.
Concept of Plane in 3 Dimensional Geometry
A first degree equation in x, y, z represents a plane in 3D
\(\begin{array}{l}ax+by+cz=0,{{z}^{2}}{{b}^{2}}+{{c}^{2}}\ne 0\end{array} \)
represents a plane.
Normal Form of a Plane
Let P be the length of the normal from the origin to the plane and l, m, n be the direction cosines of that normal. Then the equation of the plane is given by lx + my + nz = P.
Intercept form
Let a plane cut lengths a, b, and c from the coordinate axis.
Then equation of the plane is:
\(\begin{array}{l}\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\end{array} \)
Planes passing through 3 given points
Plane passing through (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is:
\(\begin{array}{l}\left| \begin{matrix} x & y & z & 1 \\ {{x}_{1}} & {{y}_{1}} & {{z}_{1}} & 1 \\ {{x}_{2}} & {{y}_{2}} & {{z}_{2}} & 1 \\ {{x}_{3}} & {{y}_{3}} & {{z}_{3}} & 1 \\ \end{matrix} \right|=0\end{array} \)
Angle between 2 planes
a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is given by
\(\begin{array}{l}\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\end{array} \)
Two sides of a plane
Consider 2 points A(x1, y1, z1) and B(x2, y2, z2) lie on the same side or opposite sides of a plane ax + by + cz + d = 0, accordingly as
\(\begin{array}{l}a{{x}_{1}}+b{{y}_{1}}+c{{z}_{1}}+d\end{array} \)
and \(\begin{array}{l}a{{x}_{2}}+b{{y}_{2}}+c{{z}_{2}}+d\end{array} \)
are of same sign or opposite sign.
Distance from a point to a plane
Distance of a point (x1, y1, z1) from a plane.
Distance of (x1, y1, z1) from ax + by + cz + d is
\(\begin{array}{l}\left| \frac{a{{x}_{1}}+b{{y}_{1}}+c{{z}_{1}}+d}{\sqrt{{{a}^{2}}+{{b}^{2}}+{{c}^{2}}}} \right|\end{array} \)
Equation of the planes bisecting the angle between 2 planes
Let a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 be 2 planes. The equation of plane bisecting the angles between them is
\(\begin{array}{l}\frac{{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z+{{d}_{1}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=\pm \frac{{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z+{{d}_{2}}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\end{array} \)
Position of origin
The origin lies in the acute or obtuse angle between a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 according as a1a2 + b1b2 + c1c2 < 0 or > 0 provided d1 and d2 are both positive.
Two Intersecting plane
If U = 0 and V = 0 be 2 planes then the plane passing through the line of their intersection is U + λV = 0λ to be determined from given condition.
Straight lines in 3D
2 intersecting planes together represent a straight line.
Equations of a straight line:
- Equation of a straight line in symmetrical form
A straight line passing through (x1, y1, z1) and having direction cosines {l, m, n} is given by
\(\begin{array}{l}\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}\end{array} \)
Equation of a straight line passing through (x1, y1, z1) and (x2, y2, z2) is
\(\begin{array}{l}\frac{x-{{x}_{1}}}{{{x}_{2}}-{{x}_{1}}}=\frac{y-{{y}_{1}}}{{{y}_{2}}-{{y}_{1}}}=\frac{z-{{z}_{1}}}{{{z}_{2}}-{{z}_{1}}}\end{array} \)
- Two plane form to symmetrical form
Let 2 planes be a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 eliminate x to get a relation between y and z. Eliminate y to get the relation between y and z. Then find in terms of x and find y in terms of z. Then equate them.
Intersection of a straight line and a plane
Let ax + by + cz + d = 0 is intersected by
\(\begin{array}{l}\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}.\end{array} \)
To find the intersection point, let
\(\begin{array}{l}\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}=t\end{array} \)
\(\begin{array}{l}\therefore x={{x}_{1}}+lt,y={{y}_{1}}+mt,z={{z}_{1}}+nt\end{array} \)
put these in the equation of plane and solve for it.
Plane through a given straight line
Let the line be
\(\begin{array}{l}\frac{x-{{x}_{1}}}{l}=\frac{y-{{y}_{1}}}{m}=\frac{z-{{z}_{1}}}{n}\end{array} \)
of the plane through this line be ax + by + cz + d = 0, then ax1 + by1 + cz1 + d = 0 and al + bm + cn = 0 and from other given conditions a, b, c are determined.
Coplanarity of Two Lines In 3D Geometry
Let 2 lines are
\(\begin{array}{l}\frac{x-{{x}_{1}}}{{{l}_{1}}}=\frac{y-{{y}_{1}}}{{{m}_{1}}}=\frac{z-{{z}_{1}}}{{{n}_{1}}}\,\,\And \,\,\frac{x-{{x}_{2}}}{{{l}_{2}}}=\frac{y-{{y}_{2}}}{{{m}_{2}}}=\frac{z-{{z}_{2}}}{{{n}_{2}}}\end{array} \)
Two lines are coplanar iff
\(\begin{array}{l}\left| \begin{matrix} {{x}_{2}}-{{x}_{1}} & {{y}_{2}}-{{y}_{1}} & {{z}_{2}}-{{z}_{1}} \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}} \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}} \\ \end{matrix} \right|=0\end{array} \)
Distance of a point from a straight line
Distance of a point from a straight line – 3D Geometry
Let the line be
\(\begin{array}{l}\frac{x-\alpha }{l}=\frac{y-\beta }{m}=\frac{z-\gamma }{n}\end{array} \)
AQ = projection of AP on the straight line
\(\begin{array}{l}=l\left( {{x}_{1}}\alpha \right)+m\left( {{y}_{1}}-\beta \right)+n\left( {{z}_{1}}-\gamma \right)\end{array} \)
\(\begin{array}{l}\therefore PQ=\sqrt{A{{P}^{2}}-A{{Q}^{2}}}\end{array} \)
Shortest distance between two skew lines
Let the 2 skew lines be
\(\begin{array}{l}\frac{x-{{x}_{1}}}{{{l}_{1}}}=\frac{y-{{y}_{1}}}{{{m}_{1}}}=\frac{z-{{z}_{1}}}{{{n}_{1}}}\end{array} \)
and \(\begin{array}{l}\frac{x-{{x}_{2}}}{{{l}_{2}}}=\frac{y-{{y}_{2}}}{{{m}_{2}}}=\frac{z-{{z}_{2}}}{{{n}_{2}}}\end{array} \)
The shortest distance is
\(\begin{array}{l}\frac{\left| \begin{matrix} {{x}_{2}}-{{x}_{1}} & {{y}_{2}}-{{y}_{1}} & {{z}_{2}}-{{z}_{1}} \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}} \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}} \\ \end{matrix} \right|}{\sqrt{\sum{{{\left( {{m}_{1}}{{n}_{2}}-{{m}_{2}}{{n}_{1}} \right)}^{2}}}}}\end{array} \)
The equation of the shortest distance is
\(\begin{array}{l}\left| \begin{matrix} x-{{x}_{1}} & y-{{y}_{1}} & z-{{z}_{1}} \\ {{l}_{1}} & {{m}_{1}} & {{n}_{1}} \\ l & m & n \\ \end{matrix} \right|=0\end{array} \)
and
\(\begin{array}{l}\left| \begin{matrix} x-{{x}_{2}} & y-{{y}_{2}} & z-{{z}_{2}} \\ {{l}_{2}} & {{m}_{2}} & {{n}_{2}} \\ l & m & n \\ \end{matrix} \right|=0\end{array} \)
Problems on 3D Geometry
Problem 1. If a variable plane forms a tetrahedron of constant volume 64 K3 with the coordinate planes then the lows of the centroid of the tetrahedron is xyz = uK3. Find u.
Answer: Let the equation of the plane be
\(\begin{array}{l}\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\end{array} \)
Centroid of the tetrahedron is (a/4, b/4, c/4).
Volume of the tetrahedron
\(\begin{array}{l}=\frac{abc}{6}=64{{K}^{3}}.\end{array} \)
So letting
\(\begin{array}{l}\frac{a}{4}=x,\frac{b}{4}=y,\frac{c}{4}=z\end{array} \)
We have
\(\begin{array}{l}\frac{abc}{6}=\frac{{{4}^{3}}xyz}{6}=64{{K}^{3}}.\end{array} \)
∴ xyz = 6K3
On comparing, we have u = 6.
Problem 2. The ration in which yz plane divides the line joining (2, 4, 5) and (3, 5, 7) is
Answer: Let the ratio be λ : 1.
x-coordinate = 0
\(\begin{array}{l}\frac{3\lambda +2}{\lambda +1} = 0\end{array} \)
\(\begin{array}{l}\lambda =-\frac{2}{3}\end{array} \)
∴ The ratio is 2:3
Problem 3. A line makes angles α, β, γ, δ with the 4 diagonals of a cube then
\(\begin{array}{l}\sum{{{\cos }^{2}}\alpha =?}\end{array} \)
Answer:
Let the direction cosine of that line be (l, m, n).
Direction cosine of 1st diagonal (1/√3, 1/√3, 1/√3).
Direction cosine of 2nd diagonal (+1/√3, -1/√3, 1/√3).
Direction cosine of 3rd diagonal (1/√3, +1/√3, -1/√3) and
Direction cosine of 4th diagonal (1/√3, -1/√3, -1/√3)
\(\begin{array}{l}\therefore \cos \alpha =\frac{l}{\sqrt{3}}+\frac{m}{\sqrt{3}}+\frac{n}{\sqrt{3}}\end{array} \)
\(\begin{array}{l}\cos \beta =\frac{l-m+n}{\sqrt{3}},\cos \gamma =\frac{+l+m-n}{\sqrt{3}},\cos \delta =\frac{l-m-n}{\sqrt{3}}\end{array} \)
\(\begin{array}{l}\therefore \sum{{{\cos }^{2}}\alpha =\frac{4\left( {{l}^{2}}+{{m}^{2}}+{{n}^{2}} \right)}{3}}=\frac{4}{3}.\end{array} \)
Problem 4. The angle between the lines
\(\begin{array}{l}\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z-2}{0}\ \text{and}\ \frac{x-1}{1}=\frac{2y+3}{3}=\frac{z+5}{2}\end{array} \)
is equal to
Answer:
\(\begin{array}{l}\cos \theta =\frac{3\times1 – 2\times3/2+0\times2}{\sqrt{3^2+(-2)^2+0^2}\sqrt{1^2+(3/2)^2+2^2} }\end{array} \)
\(\begin{array}{l}\therefore \theta =\frac{\pi }{2}\end{array} \)
Problem 5. If lines
\(\begin{array}{l}\frac{x-1}{2}=\frac{y-2}{{{x}_{1}}}=\frac{z-3}{{{x}_{2}}}\ \text{and}\ \frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\end{array} \)
lies in the same plane then for the equation \(\begin{array}{l}{{x}_{1}}{{t}^{2}}+\left( {{x}_{2}}+2 \right)t+a=0\end{array} \)
prove sum of roots = -2.
Answer: Let the plane be
\(\begin{array}{l}ax+by+cz+d=0\end{array} \)
\(\begin{array}{l}a+2b+3c+d=0\rightarrow{{}}\left( i \right)\end{array} \)
\(\begin{array}{l}2a+3b+4c+d=0\rightarrow{{}}\left( ii \right)\end{array} \)
\(\begin{array}{l}2a+{{x}_{1}}b+{{x}_{2}}c=0\rightarrow{{}}\left( iii \right)\end{array} \)
\(\begin{array}{l}3a+4b+5c=0\rightarrow{{}}\left( iv \right)\end{array} \)
\(\begin{array}{l}\text{Sum of the roots}=\frac{-\left( {{x}_{2}}+2 \right)}{{{x}_{1}}}\end{array} \)
From (i) and (ii),
a + b + c = 0
From (iii) and (iv),
a + b + c – d = 0
∴ d = 0
∴ a + 2b + 3c = 0
∴ b + 2c = 0
2a + 3b + 4c = 0
∴ b = 2c
3a + 4b + 5c = 0
2a – bc + 4c = 2c
2a = 2c
∴ a = c
equation of the plane is ax + by + cz = 0 or cx + 2cy + cz = 0
or x – 2y = z = 0
\(\begin{array}{l}\therefore {{x}_{1}}=3{{x}_{2}}=4\end{array} \)
\(\begin{array}{l}\text{Sum of roots} =\frac{-\left( {{x}_{2}}+2 \right)}{{{x}_{1}}}\end{array} \)
= -6/3
= -2
Problem 6. The line x/K = y/2 = z/-12 makes an isosceles triangle with the planes 2x + y + 3z – 1 = 0 and x + 2y – 3z – 1 = 0 then K = ?
Answer: Equation of the bisector planes are
\(\begin{array}{l}\frac{2x+y+3z-1}{\sqrt{14}}=\pm \frac{x+2y-3z-1}{\sqrt{14}}\end{array} \)
i.e. 2x + y + 3z – 1 = x + 2y – 3z – 1
or x – y + 6z = 0…..(i)
and 2x + y + 3z – 1 = -x – 2y + 3z + 1
i.e. 3x + 3y – 2 = 0 …..(ii)
so the given line must be parallel to (i) or (ii).
The coefficient of x and y of the required straight line equation is -1 and 1.
Therefore, the value of K is -2.
Problem 7. The direction cosines of normal to the plane containing lines x = y = z and x – 1 = y – 1 = (z – 1)/d are
Answer:
Problems in 3D Geometry
Let directions cosines be l,m,n.
∴ l + m + n = 0 and l + m + dn = 0
i.e. n=0
∴ l = -m from l2 + m2 + n2 = 1
We have 2l2 = 1
\(\begin{array}{l}\therefore l=\pm \frac{1}{\sqrt{2}}\end{array} \)
∴ (l, m, n) is
\(\begin{array}{l}\left( \frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0 \right)\end{array} \)
or \(\begin{array}{l}\left( -\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},0 \right).\end{array} \)
Problem 8. If the line x = y = z intersect the line x sin A + y sin B + z sin C = 2d2, x sin 2A + y sin 2B + z sin 2C = d2, where (A + B + C = π), then
\(\begin{array}{l}\sin \frac{A}{2}.\sin \frac{B}{2}.\sin \frac{C}{2}=?\end{array} \)
Answer: Let the point of intersection be (t, t, t).
\(\begin{array}{l}\therefore \left( \sin A+\sin B+\sin C \right)t=2{{d}^{2}}\end{array} \)
\(\begin{array}{l}\left( \sin 2A+\sin 2B+\sin 2C \right)t={{d}^{2}}\end{array} \)
\(\begin{array}{l}\sin 2A+\sin 2B-\sin \left( 2A+2B \right)=\frac{{{d}^{2}}}{t}.\end{array} \)
\(\begin{array}{l}\Rightarrow 2.\sin \left( A+B \right).\cos \left( A-B \right)-2\sin \left( A+B \right).\cos \left( A+B \right)=\frac{{{d}^{2}}}{t}\end{array} \)
\(\begin{array}{l}\Rightarrow 4\sin C\,\,\sin A.\,\,\sin B=\frac{{{d}^{2}}}{t.}\rightarrow{{}}\left( i \right)\end{array} \)
Again,
\(\begin{array}{l}\sin A+\sin B+\sin C\end{array} \)
\(\begin{array}{l}=2.\sin \frac{A+B}{2}.\cos \frac{A-B}{2}+2\sin \frac{C}{2}.\cos \frac{C}{2}\end{array} \)
\(\begin{array}{l}=2\cos \frac{C}{2}.\cos \frac{A-B}{2}+2\sin \frac{C}{2}.\cos \frac{C}{2}\end{array} \)
\(\begin{array}{l}=2\cos \frac{C}{2}\left( \cos \frac{A-B}{2}+\cos \frac{A+B}{2} \right)\end{array} \)
\(\begin{array}{l}=2\cos \frac{C}{2}.2cos\frac{A}{2}.\cos \frac{B}{2}=\frac{2{{d}^{2}}}{t}\rightarrow{{}}\left( ii \right)\end{array} \)
dividing (i) by (ii),
\(\begin{array}{l}\sin \frac{A}{2}.\sin \frac{B}{2}.\sin \frac{C}{2}=\frac{1}{16}.\end{array} \)
Problem 9. Equation of the sphere having center at (3, 6, -4) and touching the plane
\(\begin{array}{l}\bar{r}.\left( 2\hat{i}-2\hat{j}-\hat{k} \right)=10\ \text{is}\ {{\left( x-3 \right)}^{2}}+{{\left( y-6 \right)}^{2}}+{{\left( z+4 \right)}^{2}}={{K}^{2}},\end{array} \)
then k =
Answer: Equation of the plane is 2x – 2y – z – 10 = 0
distance from (3, 6, -4) to it is
\(\begin{array}{l}\left| \frac{6-12+4-10}{\sqrt{9}} \right|=\left| \frac{12}{3} \right|=4\end{array} \)
∴ k = 4.
Vector Algebra and 3D Geometry – Important Topics
Vector Algebra and 3D Geometry – Important Questions
Vector Algebra and 3D Geometry – Important Topics Part 2
Vector Algebra and 3D Geometry – Important Questions Part 2
3D Geometry – Top 10 Most Important and Expected JEE Questions
3-D Geometry – JEE Advanced Problems
Frequently Asked Questions
What do you mean by Direction Cosines of a line?
The direction cosine of a line is the cosine of the angle subtended by this line with the x-axis, y-axis, and z-axis respectively.
How do you represent a point in 3D Geometry?
We can represent a point in three-dimensional geometry either in cartesian form or a vector form. The cartesian form of representation of any point in 3D geometry is (x, y, z) and is with reference to the x-axis, y-axis, and z-axis respectively.
Give the formula to find the distance of a point P(x, y, z) from the origin.
The distance of a point P(x, y, z) from the origin (0, 0, 0) is given by √(x2+y2+z2).
What is the relation between direction cosines of a line?
If l, m, n denotes the direction cosines of a line, then l2+m2+n2=1.
