Formula Used
The calculator uses component formulas. Addition is A + B = (Ax + Bx, Ay + By, Az + Bz). Subtraction is A - B = (Ax - Bx, Ay - By, Az - Bz). Magnitude is |A| = sqrt(Ax² + Ay² + Az²).
The dot product is A · B = AxBx + AyBy + AzBz. The angle is θ = acos((A · B) / (|A||B|)). The cross product is A × B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx).
Projection of A on B is projB(A) = ((A · B) / |B|²)B. Rejection is A - projB(A). Unit vector A is A / |A|. Distance between vector tips is |A - B|.
How to Use This Calculator
Choose 2D or 3D mode first. Enter x, y, and z components for both vectors. In 2D mode, the z values are treated as zero. Add a scalar value if you want scaled vector results. Choose a main operation for the top summary. Then press calculate.
The result appears below the header and above the form. Review the complete table for magnitudes, unit vectors, angle, projection, rejection, distance, parallel status, and orthogonal status. Use the CSV or PDF buttons to save the same result table for homework, reports, or later checking.
Vectors Calculator Guide
A vector describes size and direction at the same time. It can show force, velocity, movement, displacement, or any quantity that points somewhere. This vectors calculator helps compare two vectors with clear component based results. You can work in two dimensions or three dimensions. In two dimensional mode, z values become zero, so ordinary plane problems stay simple.
Why Vector Components Matter
Components make vector work easier. Instead of drawing every arrow, you enter x, y, and z values. The tool then adds matching components, subtracts them, and measures length with the square root rule. This is useful in physics, engineering, graphics, surveying, navigation, and classroom math. The same approach works for small vectors and large coordinate values.
Products and Angles
The dot product tells how much two vectors point in the same direction. A positive dot product means they lean together. A negative dot product means they oppose each other. A zero dot product means the vectors are perpendicular, when neither vector has zero length. The cross product gives a new vector perpendicular to both original vectors. Its magnitude equals the area of the parallelogram formed by the two vectors.
Projection and Direction
Projection shows the part of one vector that lies along another vector. This is helpful when a force acts at an angle, or when motion must be split into useful and sideways parts. Rejection gives the remaining perpendicular part. Unit vectors are also important. They keep direction, but change length to one. Direction cosines show how strongly a vector points along each axis.
Practical Use
Use the calculator to check manual work, prepare examples, or compare vector behavior quickly. Enter exact components when possible. Select more decimal places when values are close. Always review undefined results. They usually appear when a zero vector is used in angle, unit, or projection formulas. A zero vector has no direction, so those operations cannot be measured correctly.
Before using any result in a design task, check the component units. Vectors should use the same unit system. Mixing feet with meters, or seconds with hours, can distort every output. For learning, compare the table with a hand sketch. The sketch makes direction clearer, while the numbers confirm exact size and angle. This habit reduces errors in homework, reports, and field notes too.
FAQs
What is a vector?
A vector is a quantity with magnitude and direction. It can be written with components such as (x, y) or (x, y, z). Common examples include force, velocity, and displacement.
Can this calculator work with 2D vectors?
Yes. Select 2D mode. The calculator will treat z components as zero. This lets you use the same form for plane vectors and still see cross product output.
What does the dot product show?
The dot product measures directional agreement between two vectors. It helps find angles, scalar projections, and perpendicular checks. A zero value usually means perpendicular vectors.
What does the cross product show?
The cross product creates a vector perpendicular to the two input vectors. Its magnitude equals the parallelogram area formed by those vectors.
Why is my angle not defined?
The angle is not defined when either vector has zero magnitude. A zero vector has no direction, so no meaningful angle can be measured.
What is vector projection?
Projection is the part of one vector that lies along another vector. It is useful for angled forces, shadows, components, and directional motion analysis.
How are parallel vectors detected?
The calculator checks whether the cross product magnitude is close to zero. If it is, the vectors point in the same or opposite line.
Can I export the vector results?
Yes. After calculation, use the CSV or PDF buttons. They download the result table with values and formula notes for later review.