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You are to develop a formula for the distance between these two lines, in terms of the values The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. This fact generates the vector equation of a plane: Rewriting this equation provides additional ways to describe the plane: Remember, the dot product of orthogonal vectors is zero. We say that is a normal vector, or perpendicular to the plane. Then the set of all points such that is orthogonal to forms a plane ( (Figure)). Here, we describe that concept mathematically. As you twist, the other vector spins around and sweeps out a plane. Visualize grabbing one of the vectors and twisting it. Imagine a pair of orthogonal vectors that share an initial point. Perhaps the most surprising characterization of a plane is actually the most useful. These characterizations arise naturally from the idea that a plane is determined by three points. A plane is also determined by a line and any point that does not lie on the line. For example, given two distinct, intersecting lines, there is exactly one plane containing both lines. This may be the simplest way to characterize a plane, but we can use other descriptions as well. Just as a line is determined by two points, a plane is determined by three. Similarly, given any three points that do not all lie on the same line, there is a unique plane that passes through these points. In other words, for any two distinct points, there is exactly one line that passes through those points, whether in two dimensions or three. We know that a line is determined by two points. Let be a line in space passing through point Let be a vector parallel to Then, for any point on line we know that is parallel to Thus, as we just discussed, there is a scalar, such that which gives Therefore, two nonzero vectors and are parallel if and only if for some scalar By convention, the zero vector is considered to be parallel to all vectors.Īs in two dimensions, we can describe a line in space using a point on the line and the direction of the line, or a parallel vector, which we call the direction vector ( (Figure)). If for some scalar then either and have the same direction or opposite directions so and are parallel. If two nonzero vectors, and are parallel, we claim there must be a scalar, such that If and have the same direction, simply choose If and have opposite directions, choose Note that the converse holds as well.
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Recall that parallel vectors must have the same or opposite directions. Let’s first explore what it means for two vectors to be parallel.