Chapter 12 Vectors And 3D Space Flashcards
What is the scalar product in two dimensions?
The scalar product between vectors a and b.
How is the angle between two vectors a and b expressed in three dimensions?
The angle is given by the equation a-b = |a||b| cos(θ) where θ is the angle between a and b.
What is the formula for the angle between two vectors a and b?
The angle is given by a-b = |a||b| cos(θ) where θ is the angle.
What is the vector equation of a line?
The vector equation is r = a + d, where a is the position vector of a point A on the line and d is the direction vector.
What is the Cartesian equation of a line with direction vector d?
The Cartesian equation is given by the position vector a = (d1, d2, d3) passing through point A.
How is the angle between two straight lines found?
The angle between the lines is found by calculating the scalar product d.d.
What are the four possibilities for the arrangement of two straight lines?
The lines can coincide, intersect in a single point, be parallel, or be skew.
How is the shortest distance from point P to the line r = a + Ab calculated?
The shortest distance is |(a - p) + Ab|, where l = (p - a) - b.
What are the three possibilities for the arrangement of lines in three dimensions?
The lines can be parallel, intersecting, or skew.
How can the shortest distance between two parallel lines be found?
By choosing any point on one of the lines and finding the shortest distance from that point to the second line.
What is the formula for the shortest distance between lines r1 = a + b and r2 = c + ud?
The shortest distance is (c - a) · n, where n is perpendicular to b and d (n · b = 0 and n · d = 0).
