Vector Calculus Basics for Engineering Students

Explore the basics of vectors and scalars, including magnitude and direction, notation, and properties of equal, negative, and parallel vectors.

Demonstrates vector addition and subtraction using parallelograms, showing AB + AC = AD and AB − AC via −AC, then translates vectors into coordinate form and explains scalar multiplication.

Practice scalar multiplication, addition, and subtraction of vectors using A, B, and C. See explicit calculations for three times a, sums, and differences, with upcoming vector drawings for clarity.

Learn to use position vectors from the origin to a point and compute the vector between two points by subtracting their coordinates.

Compute position vectors OA and OB from the origin for points A and B, then find AB as OB minus OA and BA as OA minus OB to show direction.

Master magnitude and unit vectors in 2d cartesian vectors, derive x and y components from cosine and sine, and use i and j notation along with core vector properties.

Learn to compute vector magnitudes for sums and differences, switch between component and Cartesian forms, and derive the vector AB, its magnitude, and the unit vector using position vectors.

Compute vector AB in Cartesian form, its magnitude and unit vector, then test parallelism via scalar multiples, and find components for vectors a, b, c, and d using given angles.

Explore 3d vectors in a Cartesian frame using the right-hand rule to locate x, y, z directions and the xy, yz, and xz planes; compute magnitudes and unit vectors.

visualize a 3d cartesian point, drop perpendiculars to the coordinate planes to locate base coordinates; compute distance, unit vectors, and magnitude relationships between points A and B.

Compute vector b of magnitude 1.5 parallel to a but opposite, using a's unit vector, then derive position vectors from transverse and azimuth angles and obtain x, y, z components.

Explore the dot product, also called the inner or scalar product, defined as a·b=|a||b|cos theta; learn its zero value for perpendicular vectors and one for parallel unit vectors.

Explore orthogonal vectors and dot products, analyze signs for acute, obtuse, and perpendicular angles, and learn direction cosines alpha, beta, gamma to find vector components and projections.

Explore dot products and their geometric meanings through 2d and 3d vector examples, determine orthogonality, angles, direction cosines, and project a onto b using unit vectors.

Master the cross product of two vectors, yielding a vector perpendicular to the plane with magnitude |a||b|sinθ, guided by the right-hand rule and determinants for scalar triple products.

Demonstrates cross product of a = i - j and b = 2j + 5k, using component and determinant methods to yield -5 i -5 j + 2 k.

Find a vector perpendicular to A and B using the cross product (determinant method). Apply cross product properties, magnitudes, and dot products with i, j, k.

Compute parallelogram area via cross product: |a × b|, triangle area 1/2|a × b|, volume a · (b × c); coplanarity when a · (b × c) = 0.

Explore 3d vectors and cross products to verify parallelograms, compute areas of parallelograms and triangles, determine volumes via triple products, and test coplanarity using dot and cross products.