Vector Algebra (Part I): A Pre-Calculus Series

Explore the triangle law of vector addition, using the head-to-tail method to obtain the resultant of two vectors and visualize how A plus B completes the triangle.

Explore vector addition using the parallelogram law and the polygon law concept to find the resultant vector as the diagonal of the parallelogram formed by two vectors.

Learn to resolve a vector into its x and y components from magnitude and angle, with a 60-degree example from the positive x axis.

Explore how to find vector components in each quadrant, determine x components from angles, and understand signs of components in the four quadrants.

Solve for the horizontal and vertical components of a vector with magnitude 8 at 27 degrees above the horizontal; the components are about 7.13 (horizontal) and 3.63 (vertical).

Compute the resultant of three forces by resolving each into x and y components, then find its magnitude and the angle it makes with the positive x-axis.

Demonstrate a vector algebra problem by calculating the x and y components, using a 45-degree angle from the negative x axis in the second quadrant, and verifying a zero resultant.

Compute the dot product, or scalar product, of two vectors. Sum the products of corresponding components and relate it to the magnitudes and the angle between them.

Explore the physical significance of the dot product and the angle between vectors, and examine how force components along a direction drive displacement in a toy car example.

Explore the properties of the dot product, including the zero vector condition, magnitude relations, and distributive and scalar multiplication rules, with applications to inequalities.

Compute the angle between two vectors using the dot product and magnitudes. Identify perpendicular vectors when the dot product is zero, and equal directions when the angle is zero.

Identify when vectors x, y, and z are perpendicular by using the dot product, recognizing zero vectors and conditions where two components vanish.

Compute the magnitudes of X plus Y and X minus Y by squaring and expanding, using the vector magnitude rules in a solved example for practice.

explains how to test the perpendicularity of vectors using dot products, highlights projection concepts, and notes that x·x is always positive, constraining vector relationships.

Determine M so that given vectors are perpendicular by setting their dot product to zero; solving gives M approximately 3.667.

Apply dot product relations and the A+B squared identity in module 3's solved example to determine an unknown component between two vectors and then verify feasible values.

Find a unit vector orthogonal to the given two-dimensional vector x = (-3, 7) using the dot product.

Compute the cross product of two vectors using a determinant, obtain the vector in i, j, k components, and relate its magnitude to the angle between the vectors.

Explore the physical significance of the cross product, its magnitude and direction relative to vectors a and b, compare it with the dot product, and apply the right-hand rule.

Learn cross product properties, including x cross x = 0, x cross y equals negative y cross x, and the distributive property of vectors.

Explore cross product applications by analyzing forces at a point, with examples from engineering, space, and mechanical devices such as doors and locks.

Explore a solved example on cross products, examining when x cross y equals zero and the resulting conditions on x, y, and their linear relationships.

Learn to find a vector perpendicular to both given vectors using the cross product, compute A x B, and obtain a unit perpendicular vector with ± direction.

Apply vector algebra to compute the cross product magnitude of vectors x and y, using magnitudes and dot product in a solved example with numerical substitution.

Learn how dot and cross products reveal vector relationships: the dot product encodes the angle and magnitude, while the cross product yields perpendicular components and maximum value at right angles.

Explore the dot and cross products, comparing their geometric meaning, magnitude relationships, and how angle governs projections, parallelism, and perpendicular components.

Explore the angle between two vectors using the Cauchy–Schwarz inequality and dot product, and learn projection and the cosine range from -1 to 1.

Explore key observations on the angle between vectors, dot and cross products, and how magnitude, direction, perpendicularity, and unit vectors shape vector operations.

Students learn to compute |A−B| by applying the identity |A−B|^2 = |A|^2 − 2 A·B + |B|^2, using |A|=|B|=1, and take the square root to get the magnitude.

solve the second example in vector algebra by determining the angle between two vectors and the magnitude of their difference using a square-based approach and a trigonometric formula.

Derive a unit vector x in three dimensions by applying magnitude, orthogonality to a given vector, and a specified angle with another vector to solve for A, B, and C.

Solve a vector algebra example: determine a vector B in a plane with x-component zero, ensure B is perpendicular to a given vector, and compute its unit form.

Module 5 presents a solved example in vector algebra, solving for the unknown q from conditions involving abc and pq vectors, and matching components to determine a, b, c.

Learn how work is done by a force by resolving it into components and multiplying by the distance traveled in that direction.

Explore how a force creates a moment or torque about a point. Relate the magnitude to the perpendicular distance to the line of action and the AP cross F.

Compute triangle area using the cross product of side vectors, applying 1/2 |AB × AC| (or AB × BC) and coordinate-based methods.

learn two methods to compute the area of a parallelogram from adjacent sides: base times height and using diagonals, and relate the area to the magnitude of the cross product.

Learn how to find the area of a parallelogram using diagonals, using the fact that diagonals bisect at the midpoint and relate triangle areas to compute the whole area.

Solve module six solved example about work done by an equal force using displacement to compute work on a particle.

Evaluate the work done by a force as a body moves from point a to point b by using the displacement vector and the force vector.

Compute the displacement vector from A to B by subtracting coordinates, derive the resulting vector, and interpret the given force data to obtain the final solution.

Compute the moment of a force about a point using the line of action and perpendicular distance AP, via the cross product, with F = (2, 3, -4) as shown.

Compute the triangle area by cross product of side vectors and half its magnitude, applying the formula with vectors such as ab and ac.

Find the area of a parallelogram using two vector methods: the cross product of adjacent sides and the diagonals, and confirm both yield the same area.