Dot product

Formula

The dot product (scalar product) of two vectors is the product of the magnitudes of the two vectors and the cosine of the angle between them. The dot product of vectors \(\vec{V}\) and \(\vec{U}\) can be calculated thanks to the following formula:

$$ \vec{V} \cdot \vec{U} = V_x.U_x + V_y.U_y + V_z.U_z = | \vec{V} |. | \vec{U} | .cos (\theta) $$

where \(\theta\) is the angle between \(\vec{V}\) and \(\vec{U}\).

Properties

If \(\vec{V}\) and \(\vec{U}\) are perpendicular, \(\vec{V} \cdot \vec{U}\) is equal to zero.
If \(\vec{V}\) is null, \(\vec{V} \cdot \vec{U}\) is equal to zero.
If \(\vec{U}\) is null, \(\vec{V} \cdot \vec{U}\) is equal to zero.
\(\vec{V} \cdot \vec{V} = | \vec{V} | ^2\)
\(\vec{V} \cdot \vec{U} = \vec{U} \cdot \vec{V}\)
\(\vec{V} \cdot (-\vec{U}) = (-\vec{V}) \cdot \vec{U} = - ( \vec{V} \cdot \vec{U} )\)

C++ source code

The following C++ function return the dot product of two vectors :

/*!
 * \brief   Compute the dot product of two vectors (this . V)
 *          The current vector is the first operand
 * \param   V is the second operand
 * \return  the dot product between the current vector and V
 */
inline double rOc_vector::dot(const rOc_vector V)
{
    return this->x()*V.x() + this->y()*V.y() + this->z()*V.z();
}

MATLAB dot product

With MATLAB, the dot product can easily be computed with the function dot(A, B) :

>> u=[1,2,3];
>> v=[4,5,6];
>> dot (u,v)

ans =

    32

See also

• Calculating the transformation between two set of points
• Catmull-Rom splines
• Check if a number is prime online
• Check if a point belongs on a line segment
• Cross product
• Common derivatives rules
• Common derivatives
• How to check if four points are coplanar
• Common integrals (primitive functions)
• Least square approximation with a second degree polynomial
• Online square root simplifyer
• Sines, cosines and tangeantes of common angles
• Singular value decomposition (SVD) of a 2×2 matrix
• Tangent line segments to circles
• Understanding covariance matrices

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