Calculating Vector Magnitude in 2D and 3D

The length or size of a 2D vector, calculated using the Pythagorean theorem.

The length or size of a 3D vector, calculated like 2D but with an extra term.

√(x² + y²) where (x, y) are the vector components.

√(x² + y² + z²) where (x, y, z) are the vector components.

Distance from origin to endpoint of the vector.

Always 1

A numerical value, without direction

A quantity with both magnitude (size) and direction, often represented by an arrow.

Vectors can be represented in component form (using coordinates) or geometrically as an arrow.

Combining vectors by adding their components or using the triangle or parallelogram rule.

Subtracting a vector is the same as adding its negative.

Multiplying a vector by a scalar changes its magnitude (size), but not its direction if the scalar is positive.

A calculation that gives a scalar value representing the component of one vector along the direction of the other.

a⋅b = |a||b|cosθ, where θ is the angle between the vectors.

(axbx + ayby + azbz) for vectors a = (ax, ay, az) and b = (bx, by, bz) in Cartesian coordinates.

Adding vectors by placing the tail of the second vector at the head of the first vector; the resultant vector starts at the tail of the first and ends at the head of the second.

Adding vectors by placing them tail-to-tail and forming a parallelogram with them as sides; the resultant vector is the diagonal starting from the common tail.

The portions of a vector along the x and y axes, determined using trigonometry or by measuring.

The order in which vectors are added does not change the resultant vector: A + B = B + A

The grouping of vectors in addition does not affect the resultant vector: (A + B) + C = A + (B + C)

The zero vector (magnitude 0) added to any vector results in the original vector: A + 0 = A

The negative of a vector, when added to the vector, results in the zero vector: A + (-A) = 0.

The length or size of a vector, representing its distance from the origin to its endpoint.

The angle a vector makes with a reference axis, usually the positive x-axis, indicating its direction.

Use the Pythagorean theorem: |v| = √(x² + y²) for 2D vectors, and |v| = √(x² + y² + z²) for 3D vectors.

Find the angle using trigonometry: θ = tan⁻¹(y/x)

Multiplying a vector by a scalar changes its magnitude, but not its direction if the scalar is positive.

Representing vectors in terms of other known vectors or components using vector addition and subtraction.

Expressing a vector as a sum of scalar multiples of other vectors.

Vectors that point in the same direction or opposite directions. Their corresponding components are proportional.

A representation of a line in space using a position vector and a direction vector. It is often given in the form r = a + tb.

A point lies on a line if its coordinates satisfy the vector equation of the line.

The angle formed by two vectors, measured in degrees or radians.

Use the dot product formula: cos(θ) = (a ⋅ b) / (|a| |b|)

Use the Pythagorean theorem: |v| = √(x² + y²) for 2D vectors, and |v| = √(x² + y² + z²) for 3D vectors.

Find the angle using trigonometry: θ = tan⁻¹(y/x).

A numerical value without direction.

Vectors that point in the same direction or opposite directions. Their corresponding components are proportional.

A representation of a line in space using a position vector and a direction vector. It's often given in the form r = a + tb.

A point lies on a line if its coordinates satisfy the vector equation of the line.

A vector with a magnitude of 1.

Divide a vector by its magnitude.