Podcast
Questions and Answers
What is the vector equation of a straight line passing through two points with position vectors a and b?
What is the vector equation of a straight line passing through two points with position vectors a and b?
- r = a + λ(b - a) (correct)
- r = a - λ(b - a)
- r = a - λ(b + a)
- r = a + λ(b + a)
Given the equation of a line in Cartesian form as z = c, what is its corresponding vector form?
Given the equation of a line in Cartesian form as z = c, what is its corresponding vector form?
- r = c(k - i)
- r = c(k + i)
- r = c(k - k1) (correct)
- r = c(k + k1)
If two lines are coplanar, what is the condition expressed by the cross product of their direction vectors?
If two lines are coplanar, what is the condition expressed by the cross product of their direction vectors?
- The cross product is normal to the lines.
- The cross product is parallel to the plane containing the lines.
- The cross product is perpendicular to the plane containing the lines. (correct)
- The cross product is tangent to the lines.
What is the vector equation of a line through the origin with direction cosines (l, m, n)?
What is the vector equation of a line through the origin with direction cosines (l, m, n)?
In the context of intersecting lines, what does the variable T represent?
In the context of intersecting lines, what does the variable T represent?
For lines given by ax + by + cz + d = 0 and a'x + b'y + c'z + d' = 0 to be coplanar, what condition must be satisfied?
For lines given by ax + by + cz + d = 0 and a'x + b'y + c'z + d' = 0 to be coplanar, what condition must be satisfied?
What is the vector equation of a straight line passing through a fixed point with position vector a and parallel to a given vector b?
What is the vector equation of a straight line passing through a fixed point with position vector a and parallel to a given vector b?
What is the equation of a line passing through the points (x1, y1, z1) and (x2, y2, z2) in vector form?
What is the equation of a line passing through the points (x1, y1, z1) and (x2, y2, z2) in vector form?
Which type of form characterizes a straight line in space by the intersection of two non-parallel planes?
Which type of form characterizes a straight line in space by the intersection of two non-parallel planes?
How is a general point on a line defined with direction ratios a, b, c and passing through the point (x1, y1, z1)?
How is a general point on a line defined with direction ratios a, b, c and passing through the point (x1, y1, z1)?
What is the equation of any plane passing through a given line in symmetrical form?
What is the equation of any plane passing through a given line in symmetrical form?
For the equation of a line in space defined by two intersecting planes, which condition must the coefficients satisfy?
For the equation of a line in space defined by two intersecting planes, which condition must the coefficients satisfy?
If a plane is parallel to the x-axis, what is the equation of the plane?
If a plane is parallel to the x-axis, what is the equation of the plane?
What is the equation of a plane cutting intercepts a, b, c on the axes?
What is the equation of a plane cutting intercepts a, b, c on the axes?
What is the vector form of the projection of a vector a on another vector b?
What is the vector form of the projection of a vector a on another vector b?
Which equation represents the yz-plane?
Which equation represents the yz-plane?
What does the equation r n ˆ d represent in the context of a plane?
What does the equation r n ˆ d represent in the context of a plane?
Which form of the projection formula in vectors involves l, m, and n?
Which form of the projection formula in vectors involves l, m, and n?
Study Notes
Vector and 3D Geometry
- Equation of a line in Cartesian form:
a1x + b1y + c1z + d1 + O(a2x + b2y + c2z + d2) = 0 - Vector equation of a straight line passing through two points with position vectors
aandb:r = a + O(b - a) - Reduction of Cartesian form of equation of a line to vector form:
x - x1 = a, y - y1 = b, z - z1 = candr = (x1, y1, z1) + O(ai + bj + ck)
Coplanar Lines
- Condition for intersection of two lines:
A / m' = B / n' = C / n - Condition of coplanarity of two lines:
aA + bm + cn sin T = 0 - Coplanarity of two lines in general asymmetric form:
ax + by + cz + d = 0anda'x + b'y + c'z + d' = 0
Projection of a Line Segment on a Line
- Projection of a line segment
PQon a line with direction cosinesl, m, n:|l(x2 - x1) + m(y2 - y1) + n(z2 - z1)| - Vector form:
a ∘ bis the projection of vectoraon another vectorb
Plane
- Intercept form:
x/a + y/b + z/c = 1 - Vector form:
r ∘ n = 0orr ∘ n = a ∘ n - Equation of a plane normal to unit vector
nand at a distancedfrom the origin:r ∘ n = d - Planes parallel to the coordinate planes:
x = 0, y = 0, z = 0 - Planes parallel to the axes:
by + cz + d = 0, bx + cz + d = 0, bx + cy + d = 0
Area of a Triangle
- Area of a triangle with vertices
A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3):|(x2 - x1)(y3 - y1) - (x3 - x1)(y2 - y1)|
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Description
Test your knowledge on vector equations and line formulas in geometry. Questions cover topics such as symmetric forms of equations, vector equations of straight lines, and determining equations of lines passing through specific points.
