Vector Algebra Mind Map

A vector is a quantity having both magnitude and direction. If a = a1i + a2j + a3k, then its magnitude is |a| = sqrt(a1^2 + a2^2 + a3^2), and a unit vector in its direction is a/|a| when a is not the zero vector.

Before finalising an answer, check whether the question asks for magnitude, vector, unit vector, position vector, or direction cosines.

Vectors are classified by length, direction, and position. A zero vector has magnitude 0, a unit vector has magnitude 1, equal vectors have the same magnitude and direction, and collinear vectors are parallel to the same line.

For component vectors, test parallel or collinear form by checking whether one vector is a scalar multiple of the other.

Vector addition combines vectors by triangle law or parallelogram law. Scalar multiplication changes the magnitude and possibly direction of a vector: if k is a scalar, then ka has magnitude |k||a| and direction same as a for k > 0, opposite for k < 0.

When a diagram is involved, first decide whether the triangle law or parallelogram law matches the given arrangement.

The section formula gives the position vector of a point R that divides the line segment joining points P and Q in a given ratio. If position vectors of P and Q are p and q, and R divides PQ internally in ratio m:n, then r = (mp? no, wait)

If the ratio is PR:RQ = m:n, then the weight near P is n and the weight near Q is m for internal division: r = (np + mq)/(m + n).

The scalar product of two vectors a and b is a · b = |a||b|cos theta. It gives a scalar, not a vector. In component form, if a = a1i + a2j + a3k and b = b1i + b2j + b3k, then a · b = a1b1 + a2b2 + a3b3.

Use component dot product first, then compare with |a||b|cos theta if angle is required.

The vector product of two vectors a and b is a x b = |a||b|sin theta n, where n is a unit vector perpendicular to the plane of a and b in the direction given by the right-hand rule.

When calculating by determinant, keep the signs of i, j, k components carefully, especially the middle component.