The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straight line parallel to . It is a vector parallel to , defined as: where is a scalar, called the scalar projection of onto , and is the unit vector in the direction of . In turn, the scalar projection is defined as: where the operator ⋅ denotes a dot product, ‖a‖ is the length of , and θ is the angle between and . Which finally gives: The scalar projection is equal to the length of the vector projection, with a minus sign if the direction of the projection is opposite to the direction of . The vector component or vector resolute of perpendicular to , sometimes also called the vector rejection of from (denoted ), is the orthogonal projection of onto the plane (or, in general, hyperplane) orthogonal to . Both the projection and rejection of a vector are vectors, and their sum is equal to , which implies that the rejection is given by: Notation Typically, a vector projection is denoted in a bold font (e.g. ), and the corresponding scalar projection with normal font (e.g. a1). In some cases, especially in handwriting, the vector projection is also denoted using a diacritic above or below the letter (e.g., or a1). The vector projection of on and the corresponding rejection are sometimes denoted by and , respectively. Definitions based on angle θ Scalar projection The scalar projection of on is a scalar equal to where θ is the angle between and . A scalar projection can be used as a scale factor to compute the corresponding vector projection. Vector projection The vector projection of on is a vector whose magnitude is the scalar projection of on with the same direction as . Namely, it is defined as where is the corresponding scalar projection, as defined above, and is the unit vector with the same direction as : Vector rejection By definition, the vector rejection of on is: Hence, Definitions in terms of a and b When is not known, the cosine of can be computed in terms of and , by the following property of the dot product Scalar projection By the above-mentioned property of the dot product, the definition of