Vectors in 3D
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Vectors
Drag the glowing tips, or type values. Angles are measured from the x axis, and θ is the angle from A to B. Pick a result card to see it built in the picture.
Vector A of size 3 and vector B of size 4 with 90 degrees between them. The resultant is 5, 53.13 degrees from A. The dot product is 0 and the cross product has size 12.
Show the formulas
- Resultant
- R = √(A² + B² + 2AB cos θ) = √(3² + 4² + 2 × 3 × 4 × cos 90°) = 5
- Direction
- tan β = B sin θ / (A + B cos θ) = 4 × sin 90° / (3 + 4 × cos 90°), so β = 53.13° from A
- Difference
- |A - B| = √(A² + B² - 2AB cos θ) = √(3² + 4² - 2 × 3 × 4 × cos 90°) = 5
- Dot product
- A·B = AB cos θ = 3 × 4 × cos 90° = 0
- Cross product
- |A×B| = AB sin θ = 3 × 4 × sin 90° = 12
- Components
- A·B = AxBx + AyBy + AzBz = 3 × 0 + 0 × 4 + 0 × 0 = 0
- Determinant
- A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 12k
- Unit vector
- Â = A / |A| = 1i
Pick a case
- Resultant of two vectors
- Resultant at 90 degrees
- Resultant at 60 degrees
- Resultant at 120 degrees
- Resultant at 180 degrees
- Resultant at 0 degrees
- Resultant of two equal vectors
- Maximum and minimum resultant
- Triangle law of vector addition
- Parallelogram law of vector addition
- Vector subtraction
- Dot product of two vectors
- Cross product of two vectors
- Angle between two vectors
- Unit vector
- Rectangular components
- Components on an inclined plane
- Polygon law of vectors
Same chapter: dimensional formula, unit converter, significant figures, error analysis, mirrors and lenses, projectile motion, atoms and quanta, circuits.
Adding and multiplying vectors
How do I find the resultant of two vectors?
Put A and B tail to tail with the angle θ between them. The resultant has magnitude R = √(A² + B² + 2AB cos θ) and makes an angle β with A where tan β = B sin θ / (A + B cos θ).
What is the difference between the dot product and the cross product?
The dot product A·B = AB cos θ is a scalar: it is large when the vectors point the same way. The cross product A×B has magnitude AB sin θ and is a vector at right angles to both, found with the right-hand rule.
How is the direction of a cross product found?
Curl the fingers of your right hand from A towards B through the smaller angle. Your thumb points along A×B. Swapping the order reverses it, so B×A = -(A×B).
What is a unit vector?
A vector of magnitude 1 that only gives a direction. For any vector A, the unit vector is  = A / |A|. In components, i, j and k are the unit vectors along x, y and z.
Why does the 3D picture show the cross product rising out of the plane?
A×B is perpendicular to the plane that holds A and B. With A and B lying flat, the product points straight up or down, which is the +z or -z direction.