Vector

High School

Definition

A line with an arrowhead at its end in mathematics that has both a direction and a magnitude (a size) associated with it. The length of the arrow can tell you the magnitude and the arrowhead shows you the direction.

Worked examples

\(\vec{v} = \langle 3, 4 \rangle\)

This vector points 3 units right and 4 units up; its magnitude is \(\sqrt{3^2 + 4^2} = 5\).

\(\vec{AB}\) from A(1, 2) to B(4, 6) is \(\langle 3, 4 \rangle\)

Subtract coordinates: the vector points from start to end with direction and length.

Common mistakes

\(\vec{v} = \langle 3, 4 \rangle\) has magnitude \(3 + 4 = 7\) → \(|\vec{v}| = \sqrt{3^2 + 4^2} = 5\) Use the Pythagorean theorem, not simple addition, to find magnitude.
Drawing the arrow anywhere — position doesn't matter → In position vectors, the tail is at the origin; in displacement, start point matters Context determines whether you care about position or just direction and magnitude.
\(\vec{u} + \vec{v}\) means multiplying their magnitudes → \(\vec{u} + \vec{v}\) means adding corresponding components Vector addition is component-wise, not scalar multiplication.

Where you'll use it next

Vectors are foundational for physics (force, velocity), vector operations and dot/cross products in precalculus, parametric equations, multivariable calculus, and linear algebra.

Found in 1 StudyPug lesson

Mastering Vectors: From Basics to Advanced Applications

Grade 12 Math

Dive into the world of vectors! Learn how to represent, manipulate, and apply vector concepts in various fields. Boost your problem-solving skills and prepare for advanced math and physics topics.

12th Grade12th