Vector Equations and Line Formulas Quiz

18 Questions

What is the vector equation of a straight line passing through two points with position vectors a and b?

r = a + λ(b - a)

Given the equation of a line in Cartesian form as z = c, what is its corresponding vector form?

r = c(k - k1)

If two lines are coplanar, what is the condition expressed by the cross product of their direction vectors?

The cross product is perpendicular to the plane containing the lines.

What is the vector equation of a line through the origin with direction cosines (l, m, n)?

r = λ(l i + m j + n k)

In the context of intersecting lines, what does the variable T represent?

Angle between the lines

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?

$|a b c| = |a' b' c'|$

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?

r = a + λb

What is the equation of a line passing through the points (x1, y1, z1) and (x2, y2, z2) in vector form?

(x - x2) / (x1 - x2) = (y - y1) / (y2 - y1) = (z - z2) / (z1 - z2)

Which type of form characterizes a straight line in space by the intersection of two non-parallel planes?

Non-symmetric form

How is a general point on a line defined with direction ratios a, b, c and passing through the point (x1, y1, z1)?

(x1 + ar, y1 + br, z1 + cr)

What is the equation of any plane passing through a given line in symmetrical form?

(r ⋅ n₂) = d₁

For the equation of a line in space defined by two intersecting planes, which condition must the coefficients satisfy?

|a₁b₂ − a₂b₁| ≠ 0

If a plane is parallel to the x-axis, what is the equation of the plane?

by + cz + d = 0

What is the equation of a plane cutting intercepts a, b, c on the axes?

a/x + b/y + c/z = 1

What is the vector form of the projection of a vector a on another vector b?

|a| (l î + m ĵ + n k̂)

Which equation represents the yz-plane?

x = 0

What does the equation r n ˆ d represent in the context of a plane?

Vector equation of a plane normal to unit vector n at distance d from the origin

Which form of the projection formula in vectors involves l, m, and n?

| l (x2 – x1) + m(y2 – y1) + n(z2 – z1) |

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 a and b: 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 = c and r = (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 = 0 and a'x + b'y + c'z + d' = 0

Projection of a Line Segment on a Line

Projection of a line segment PQ on a line with direction cosines l, m, n: |l(x2 - x1) + m(y2 - y1) + n(z2 - z1)|
Vector form: a ∘ b is the projection of vector a on another vector b

Plane

Intercept form: x/a + y/b + z/c = 1
Vector form: r ∘ n = 0 or r ∘ n = a ∘ n
Equation of a plane normal to unit vector n and at a distance d from 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)|

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.

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