Calculus 3 with the Math Sorcerer

Compute the dot product of two space vectors by multiplying corresponding coordinates and summing the results, illustrated with vectors [1, 2, 4] and [-3, -2, 0], yielding -7.

Compute the angle between space vectors using the dot product, confirming orthogonality when u·v = 0 and theta = pi/2 (90 degrees) through component form and the cosine formula.

Derive direction angles and direction cosines for a vector in 3-D, normalize it, express V in unit-vector components, and verify cos^2 alpha plus cos^2 beta plus cos^2 gamma equals 1.

Find the direction angles of the vector from components (5, 9, 7) by computing magnitude sqrt(155) and the cosines 5/sqrt(155), 9/sqrt(155), 7/sqrt(155), then determine alpha, beta, gamma in degrees.

Compute the cross product of two vectors by expanding the determinant with i, j, k and 2x2 minors.

Compute the cross product of two vectors using a determinant, express the result in component form, and verify the resulting vector is perpendicular to both originals.

Learn to calculate the distance between two points in space using the 3d distance formula, with a step-by-step example that yields the result sqrt(6).

derive a line in space from a point and a parallel vector; obtain parametric form x1 + a t, y1 + b t, z1 + c t, and symmetric form.

Derive the plane equation from a point on the plane and a normal vector, using n · (x - x1, y - y1, z - z1) = 0.

Learn to find the equation of a plane through a point with a given normal vector using the form (x−x0)A+(y−y0)B+(z−z0)C=0, resulting in y=5 for the example.

Find the equation of a plane from a point and a normal vector using the standard form a(x−x1)+b(y−y1)+c(z−z1)=0, with the normal components a,b,c and the point (x1,y1,z1).