Two vectors and are called parallel if they are simply scalar multiples of one another, so for some nonzero number . As this number can be either positive or negative, vectors which point in the same direction and vectors which point in exactly opposite directions are considered parallel.

An equation of the plane containing the point (x0,y0,z0) with normal vector N = is A(x – x0) + B(y – y0) + C(z – z0)=0. Note: The equation of any plane can be expressed as Ax + By + Cz = D. This is called the standard form of the equation of a plane.

In Figure 5, the faces of the rectangular box are formed by the three coordinate planes x = 0 (the yz-plane), y = 0 (the xz-plane), and z = 0 (the xy-plane), and the planes x = a, y = b, and z = c.

A vector having magnitude of 1 is known as the unit vector. As its magnitude is 1, they are also known as the direction vectors. A unit vector is denoted as →a=x∧i+y∧j+z∧k. Here, as we have to find the unit vectors in XY plane, the z coordinate will be 0.

Standard equations for some special planes. 0(x – 0) + 0(y – 0) + 1(z – 0) = 0 , i.e. z = 0. Similarly, the y-z-plane has standard equation x = 0 and the x-z-plane has standard equation y = 0. A plane parallel to the x-y-plane must have a standard equation z = d for some d, since it has normal vector k.

The cross product property (“cross multiplication”) is a way to eliminate fractions in an equation. Simply multiply each numerator by the other side’s denominator, get rid of the actual denominators, and solve.

Four primary uses of the cross product are to: 1) calculate the angle ( ) between two vectors, 2) determine a vector normal to a plane, 3) calculate the moment of a force about a point, and 4) calculate the moment of a force about a line.

The Cross Product gives a vector answer, and is sometimes called the vector product. But there is also the Dot Product which gives a scalar (ordinary number) answer, and is sometimes called the scalar product.

Using the cross product to find the angle between two vectors in R3. Let u=⟨1,−2,3⟩andv=⟨−4,5,6⟩. Find the angle between u and v, first by using the dot product and then using the cross product. I used the formula: U⋅V=||u||||v||cosΔ and got 83∘ from the dot product.

The Scalar Method in 3 Dimensions For moments in three dimensions, the moment vector will always be perpendicular to both the force vector F and the distance vector d. If you do this, your thumb should point out along the line in the direction of the moment vector.