Vectors application checklist: Lines and planes in 3D space (H2 math)

Vectors application checklist: Lines and planes in 3D space (H2 math)

Vector geometry checklist: Avoiding common mistakes in H2 math .

Check our other pages :

Frequently Asked Questions

The vector equation of a line is **r** = **a** + λ**d**, where **a** is a position vector of a point on the line, **d** is the direction vector of the line, and λ is a scalar parameter. You can find it by identifying a point on the line and a vector parallel to the line.
First, check if their direction vectors are parallel (scalar multiples of each other). If yes, they are either parallel or coincident. If not, check if they intersect by equating their vector equations and solving for the parameters. If no solution exists, and the lines are not parallel, they are skew.
The equation of a plane can be expressed in vector form as **r** ⋅ **n** = **a** ⋅ **n**, where **r** is the position vector of a general point on the plane, **n** is a normal vector to the plane, and **a** is the position vector of a known point on the plane. The Cartesian form is ax + by + cz = d.
The normal vector can be found from the coefficients of x, y, and z in the Cartesian equation of the plane (ax + by + cz = d), which gives **n** = (a, b, c). Alternatively, if you have three points on the plane, you can find two vectors lying in the plane and take their cross product.
The distance from a point P with position vector **p** to a plane **r** ⋅ **n** = d is given by the formula |(**p** ⋅ **n** - d)| / |**n**|.
To find the line of intersection, solve the system of equations formed by the Cartesian equations of the two planes. Express one variable in terms of another (e.g., z in terms of x), and then substitute back into one of the equations to express another variable in terms of the same variable (e.g., y in terms of x). Finally, express the solution in vector form.
The angle between two planes is the angle between their normal vectors. If **n1** and **n2** are the normal vectors, the angle θ is given by cos θ = (**n1** ⋅ **n2**) / (|**n1**| |**n2**|).
The angle between a line and a plane is the complement of the angle between the lines direction vector and the planes normal vector. If **d** is the direction vector of the line and **n** is the normal vector of the plane, the angle θ is given by sin θ = (**d** ⋅ **n**) / (|**d**| |**n**|).
The shortest distance between two skew lines is the length of the common perpendicular. It can be found using the formula |(**a2** - **a1**) ⋅ (**d1** × **d2**)| / |**d1** × **d2**|, where **a1** and **a2** are position vectors of points on the lines, and **d1** and **d2** are their direction vectors.
Substitute the coordinates of the point into the Cartesian equation of the plane. If the equation is satisfied, the point lies on the plane. Alternatively, substitute the position vector of the point into the vector equation of the plane and check if the equation holds.