Vectors in Space!!! Cool stuff in Section 8.6b

Just as in planes, sets of equivalent directed line segments in space are called vectors. Denoted with ordered triples: Zero Vector: Standard Unit Vectors: Vector v can be represented with these standard unit vectors:

Let’s see this graphically: z y x k i j (0, 0, 1) (0, 1, 0) (1, 0, 0) v

Vector Relationships in Space How do we write the vector v defined by the directed line segment from P to Q? Multiplying a vector by a scalar?

For vectors: Equality:v = w if and only if: Addition:v + w = Subtraction:v – w = Magnitude:|v| = Dot Product:v w = Unit Vector:, is the unit vector in the direction of v.

Computing with Vectors

A jet airplane just after takeoff is pointed due east. Its air velocity vector makes an angle of 30 with flat ground with an airspeed of 250 mph. If the wind is out of the southeast at 32 mph, calculate a vector that represents the plane’s velocity relative to the point of takeoff. Let i point east, j point north, and k point up. A diagram? Plane’s air velocity: a = Wind velocity (pointed northwest): w =

A jet airplane just after takeoff is pointed due east. Its air velocity vector makes an angle of 30 with flat ground with an airspeed of 250 mph. If the wind is out of the southeast at 32 mph, calculate a vector that represents the plane’s velocity relative to the point of takeoff. Let i point east, j point north, and k point up.A diagram? Velocity relative to the ground is v = a + w: v = = i j + 125k

A rocket soon after takeoff is headed east and is climbing at a 80angle relative to flat ground with an airspeed of 12,000 mph. If the wind is out of the southwest at 8 mph, calculate a vector v that represents the rocket’s velocity relative to the point of takeoff. Let i point east, j point north, and k point up.A diagram? Rocket’s velocity relative to the air: Air’s velocity relative to the ground: Rocket’s velocity relative to the ground: v = i j + 11, k

Lines in Space First-degree equations in three variables graph as… Planes!!! So how do we get lines?  The pair of equations y = 0 and z = 0 give…the x-axis!!! For more complicated lines, we can use: one vector equation, or a set of three parametric equations.

Suppose L is a line through the point for some real number t. The vector v is a direction vector for line L. and in the direction of a nonzero vector v = Then for any point on L, x y z L

Let: x y z L Then The vector equation of line L: In component form:

Let: x y z L Then This can be expressed as the parametric equations:

If L is a line through the point Equations for a Line in Space in the direction of a nonzero vector v =, then a point is on L if and only if Vector Form:where OR Parametric Form: where t is a real number.

The line through Practice Problems with direction vector can be written: in vector form as or in parametric form as How can we “see” this graphically???

Using the standard unit vectors i, j, and k, write a vector equation for the line containing the points A(3, 0, –2) and B(–1, 2, –5), and compare it to the parametric equations for the line. Practice Problems The line is in the direction of So using, the vector equation of the line becomes

Using the standard unit vectors i, j, and k, write a vector equation for the line containing the points A(3, 0, –2) and B(–1, 2, –5), and compare it to the parametric equations for the line. Practice Problems The parametric equations are the three component equations: