Taking a scalar product of two vectors results in a number , as its name indicates. Scalar products are used to define work and energy relations. For example, the work that a force performs on an object mshs healthnet while causing its displacement is defined as a scalar product of the force vector with the displacement vector. A quite different kind of multiplication is a vector multiplication of vectors.

The direction of the cross product is given by the progression of the corkscrew. The dot product essentially tells us how much of the force vector is applied in the direction of the motion vector. The dot product can also help us measure the angle formed by a pair of vectors and the position of a vector relative to the coordinate axes.

Note that the torque vector is orthogonal to both the force vector and the radius vector. We have found the components of a vector given its initial and terminal points. In some cases, we may only have the magnitude and direction of a vector, not the points. For these vectors, we can identify the horizontal and vertical components using trigonometry (Figure 2.15). Unlike vectors, scalars are generally considered to have a magnitude only, but no direction.

In three dimensions, the equations describe planes that are parallel to the coordinate planes. As described earlier, vectors in three dimensions behave in the same way as vectors in a plane. The geometric interpretation of vector addition, for example, is the same in both two- and three-dimensional space (). Let and Express in component form and in standard unit form. The equation of a circle is derived using the distance formula in two dimensions. In the same way, the equation of a sphere is based on the three-dimensional formula for distance.

Understand multiplying a vector by a scalar and by another vector. Learn scalar multiplication of vectors with an overview of what scalar and vector quantities are. If there’s something you don’t know, your very first impulse should be to find out how it is defined. Math books are careful to define the terms they use very precisely.


Hence, proceeding on the lines of the above theorem, we have the following result. The next result gives the proof of the QR decomposition for real matrices. The readers are advised to prove similar results for matrices with complex entries. This decomposition and its generalizations are helpful in the numerical calculations related with eigenvalue problems . We now prove the most important initial result of this section. A very useful and a fundamental inequality, commonly called the Cauchy-Schwartz inequality, concerning the inner product is proved next.

Moreover, this tool is useful for checking if two angles are coterminal. Divide the resultant by the magnitude of the second vector. Divide the dot product by the magnitude of the first vector.

There is another form of the formula that we used to get the directional derivative that is a little nicer and somewhat more compact. It is also a much more general formula that will encompass both of the formulas above. In this case let’s first check to see if the direction vector is a unit vector or not and if it isn’t convert it into one. To do this all we need to do is compute its magnitude.

Vectors which are perpendicular are called orthogonal; if they are also of unit length, they are referred to as orthonormal . ■Two vectors are orthogonal if and only if their dot product is zero. ■The commutative and distributive laws hold for the dot product of vectors in ℝn. The final property of the dot product that we will discuss comes from geometry.