Dot Product Calculator
The angle between two vectors, in any dimension.
Work out Dot Product. The angle between two vectors, in any dimension. Shows the working, not just the answer.
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Dot product
32
12.93° between them · Acute angle — they broadly agree in direction
The dot product multiplies matching components and adds them, which equals |A||B|cos θ — so it encodes the angle between two vectors in a single number, in any number of dimensions. Zero means perpendicular, and that is the property doing most of the work in practice: it is how you test orthogonality, how you project one vector onto another, and how a great deal of computer graphics decides whether a surface faces the light. The projection row is the length of A's shadow on B. It is negative when the angle is obtuse, which is meaningful rather than an error — the shadow falls the other way.
How the Dot Product Calculator works
Dot product, magnitudes, the angle between two vectors and the projection of one onto the other. The dot product equals |A||B|cos θ, so it encodes the angle in a single number — in any number of dimensions.
Also known as: angle between two vectors · are these vectors perpendicular · scalar product calculator · project one vector onto another
Frequently asked questions
What does the dot product mean?
It multiplies matching components and adds them, which works out to |A||B|cos θ. So it is large when two vectors point the same way, zero when they are perpendicular, and negative when they oppose.
How do I test if two vectors are perpendicular?
Their dot product is zero. That test is the reason the dot product appears everywhere from orthogonal bases to collision detection in graphics.
What is a scalar projection?
The length of one vector's shadow on another — how much of A points along B. It is negative when the angle is obtuse, which is meaningful: the shadow falls the other way.
Does the dot product work in more than three dimensions?
Yes, in any number. Unlike the cross product, which only exists in three dimensions, the dot product is defined for vectors of any matching length and is used routinely in high-dimensional data work.
What is the difference between dot and cross products?
The dot product returns a number and measures alignment; the cross product returns a vector perpendicular to both and measures the area they span. One uses cosine, the other sine.
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