Vectors in Space

Vectors describe motion in three-dimensional space with both size and direction. They are useful for quantities such as displacement and velocity, since those measurements need more than just a number. A vector is shown as an arrow from one point to another. The arrow’s length gives the magnitude, and its direction shows the way the motion points.

In three dimensions, a vector is written in terms of its x-, y-, and z-components. These components are often shown with angle brackets, such as ⟨x,...

Cable tension and force balance are shown with a steel beam held by two identical cables. The beam has a downward weight of 5000 N. The cables support it from opposite sides. Because the setup is symmetric, each cable makes a 60° angle with the horizontal beam and carries the same tension.

The beam stays completely at rest, so it is in static equilibrium. Static equilibrium means the total horizontal force and the total vertical force are both zero. Each cable pulls in its own direction, so...

The dot product connects vectors, angles, and projections in vector algebra. It combines two vectors to produce a scalar, which is a single number. This makes it useful in math and in physics, especially for work.

For vectors a = ⟨a1, a2, a3⟩ and b = ⟨b1, b2, b3⟩, the dot product is found by multiplying matching components and adding them: a · b = a1b1 + a2b2 + a3b3. The operation is commutative, so the order does not matter. It is also distributive over vector addition and works with scalar...

The cross product connects vector direction, area, and rotation in three-dimensional space. When two non-zero vectors are not parallel, they define a unique plane and form a parallelogram. Their cross product gives a third vector that is perpendicular to that plane.

The direction of this new vector is found with the Right-Hand Rule. Point the fingers of your right hand along the first vector and curl them toward the second vector. Your thumb shows the direction of the result. This convention...

A line in three-dimensional space can be described with a vector equation. This method uses a known point on the line and a direction vector. It is a useful way to model straight paths in space.

The transcript uses a surveying laser beam as an example. A laser beam from a surveying instrument can be treated as a line across a construction site. The beam starts at a known point, called P₀, whose position vector is r₀. The beam also has a direction vector v, which is parallel to its path.

Any...

A plane in three-dimensional space can be described by one point on the plane and a normal vector. The normal vector is perpendicular to the surface, and it sets the plane’s orientation. In geometry, this gives a clear way to model a flat surface in space.

The transcript uses a sloped glass panel on a building façade as a real-world example. Let r₀ be the position vector of a known point on the panel, such as the base mounting point. Let n be the normal vector to the panel. Any other point r...

Cylindrical surfaces in three-dimensional space come from moving a two-dimensional curve in a straight line. As the curve slides, it leaves behind translated copies that build a surface. These copies form parallel rulings, which are straight lines in the same fixed direction.

This idea helps describe some 3D shapes with simple equations in Cartesian coordinates. A cylindrical surface is often shown by an equation that leaves out one of the three variables. For example, y = x^2 does not include...

Quadric surfaces are smooth three-dimensional shapes described by second-degree equations in x, y, and z. The way the squared and linear terms are arranged determines the surface. The main quadric surfaces are ellipsoids, cones, paraboloids, and hyperboloids.

An ellipsoid is a closed surface that appears when all three variables are squared with the same sign. It is bounded and symmetric about its principal axes. Cross-sections taken along those axes produce circles or ellipses. A standard...

Crosswind vector navigation in 2D shows how wind changes an airplane’s path. A plane flew due north at 180 km/h in still air, but after 30 minutes it was 80 km off course. Its track was about 5 degrees east of north, which shows that a crosswind pushed it away from the intended route.

The plane’s ground path was diagonal, not straight north. That means its effective ground speed was reduced to 160 km/h and shifted slightly east. By studying the displacement from the planned path, the wind’s...

Three-dimensional static equilibrium can be used to find the tension in cables that support a chandelier. In this setup, the chandelier weighs 1000 N and sits at the origin of a 3D coordinate system. Three ceiling anchor points are located above it, and each cable runs from the chandelier to one anchor point.

To analyze the forces, the direction of each cable must first be described with a position vector. A position vector is a vector that points from the chandelier to an anchor point. These...