Vector Magnitude, Addition & Dot Product Calculator

Vector Guide

What's the key thing to understand about Vector Magnitude and Unit Vectors?

For vector a = (x, y): |a| = √(x² + y²). For 3D a = (x, y, z): |a| = √(x² + y² + z²). Unit vector â = a / |a| — a vector of magnitude 1 in the direction of a. Example: a = (3, 4). |a| = √(9+16) = 5. â = (3/5, 4/5) = (0.6, 0.8). Check: |â| = √(0.36 + 0.64) = 1 ✓. Standard basis vectors: i = (1,0,0), j = (0,1,0), k = (0,0,1). Any 3D vector: a = a_x i + a_y j + a_z k.

What do I need to know about Vector Addition and Subtraction?

Component-wise: (a_x + b_x, a_y + b_y). Geometrically: tip-to-tail addition (place vectors head to tail). Parallelogram law: resultant is the diagonal. Example: a=(3,4), b=(1,2). a+b = (4,6). |a+b| = √(16+36) = √52 = 7.21. Triangle of forces: three forces in equilibrium form a closed triangle. Resolution of forces: any force can be resolved into perpendicular components: F_horizontal = F cosθ, F_vertical = F sinθ. This is used constantly in mechanics problems.

What do I need to know about Dot Product?

a·b = a_x b_x + a_y b_y + a_z b_z = |a||b|cos θ. If a·b = 0: vectors are perpendicular (cos 90° = 0). If a·b > 0: angle < 90°. If a·b < 0: angle > 90°. Finding angle: θ = arccos(a·b / (|a||b|)). Example: a=(3,4), b=(1,2). a·b = 3+8 = 11. |a|=5, |b|=√5=2.236. cos θ = 11/(5×2.236) = 0.985. θ = arccos(0.985) = 9.93°. Work done: W = F·d = |F||d|cos θ. Only the component of force in the direction of motion does work.

What's the key thing to understand about Cross Product?

a × b = (a_y b_z − a_z b_y, a_z b_x − a_x b_z, a_x b_y − a_y b_x). |a × b| = |a||b|sin θ. Direction: perpendicular to both a and b (right-hand rule). If a × b = 0: vectors are parallel. Applications: torque τ = r × F. Angular momentum L = r × p. Area of parallelogram formed by two vectors = |a × b|. Normal to a surface: cross product of two vectors in the plane gives the surface normal. The cross product only exists for 3D vectors — no 2D equivalent.