Vector product form of a line

Vector Product Form of a Line

Vector Form of a Line

• A line in three-dimensional space can be specified in the vector form.
• The vector form of a line is r = a + λb, where a is a point on the line, b is a vector parallel to the line, λ is a scalar parameter, and r is the position vector of any point on the line.
• The two vectors a and b define the line and these vectors can be written as ai + bj + ck where i,j and k are the standard unit vectors.

Interpreting the Vector Form

• a or ‘position vector of a fixed point’ is the position vector of a point A on the line.
• b or ‘direction vector of the line’ gives the direction of the line, which can be either direction along the line.
• r represents the position vector of any general point P on the line.
• λ is a scalar quantity, as λ varies, P moves along the line.

Points on the Line

• Any point on the line can be found by changing the value of λ.
• The point A corresponds to λ = 0.
• If λ increases, we move in the direction of b. If λ decreases, we move in the opposite direction.

Equating Coefficients

• Equations of lines in vector form can be converted to Cartesian form by equating coefficients of the corresponding components i, j, k.

Parallel Lines

• Two lines are parallel if their direction vectors are proportional i.e. identical or one is a scalar multiple of the other.

Intersection of Line

• If two lines intersect, the position vectors of their points of intersection must satisfy both equations of the lines. To find the intersection, equate the coefficients and solve the resulting equations.

Perpendicular Lines

• Two lines are perpendicular if the dot product of their direction vectors equals zero.

Remember, the vector form of a line, r = a + λb, is important in many areas of Further Pure Mathematics, so be sure to understand thoroughly how it is derived, interpreted and used in problems.