A Mathematical Blog

In math and physics we often have to deal with vectors in component form. \(\vec{u} = 2\hat{i} - 3\hat{j}\) This is a vector in component form. \(2\hat{i}\) and \(-3\hat{j}\) are its vector components , while the multiples of \(\hat{i}\) and \(\vec{j}\), i.e., 2 & -3, are its scalar components .  Expressing a vector in component form basically means writing that vector in terms of some “base” vectors like \(\hat{i}\) and \(\hat{j}\).  \(\hat{i}\) is a vector of length one unit in the positive x-direction; \(\hat{j}\) is a vector of length one unit in the positive y-direction. Let’s say vector u’s tail is at the origin. Starting from the origin, we travel two times the length of \(\hat{i}\)\((2 \times 1 = 2)\) in the positive x-direction, and then travel 3 times the length of \(\hat{j}\)\((3 \times 1 = 3)\) in the negative y-direction(because the vector component is negative). We have reached the head of the vector u.  So when a vector's tail is at the origin...

Question :  Find the volume of parallelepiped spanned by (non-coplanar) vectors : \(\vec{a} = 1\hat{i} + 1\hat{j} + 4\hat{k}\), \(\vec{b} = 2\hat{i} + 3\hat{j} - 1\hat{k}\) and \(\vec{c} = -1\hat{i} + 1\hat{j} + 3\hat{k}\). Solution :  “Spanned by vectors” here means that the vectors are along the three adjacent sides of the parallelepiped. They have magnitudes equal to the lengths of the corresponding edges. For finding the volume of parallelepiped, first we want to select its base. Any face can be selected as its base.  For the chosen base in the figure, \(\vec{b}\) and \(\vec{c}\) are the vectors along its adjacent sides(shown separately). So for the volume, we first find \(\vec{b} \times \vec{c}\)… and ‘dot’ the result  with \(\vec{a}\). Volume of parallelepiped \(= |\vec{a} \cdot (\vec{b} \times \vec{c})|\) If a different base was selected, say the one facing us in the figure with adjacent sides as \(\vec{a}\) and \(\vec{b}\), then the volume will...

Another application of cross product of vectors occurs in geometry, or rather application of both the dot and the cross product, wherein it helps with finding the volume of a solid called parallelepiped. A parallelepiped is a six faced three dimensional shape. All six faces are parallelograms(hence the name). The opposite faces are parallel and congruent(same shape and area).  Firstly, by looking at the figure I hope that we all agree that the volume of parallelepiped is the area of the base times the height, where any face of the solid can be chosen as its base. The height is the length of the perpendicular from the chosen base to the opposite face. Volume of parallelepiped = Area of PQRS \(\times h\) Take any vertex and define three vectors, \(\vec{a}, \vec{b}\) and \(\vec{c}\), from that vertex along the three edges, as shown. Their magnitude equals the length of the corresponding edge. Now, the base of the solid is a parallelogram. In my previous post, we saw that the...

The cross product of two vectors can be determined in any of the following two ways : \(\vec{a} \times \vec{b} = \Biggl|\matrix{\hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3}}\Biggl|\)                                            where \(\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}\) and,              \(\vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}\). \(|\vec{a} \times \vec{b}| = |\vec{b} \times \vec{a}| = |\vec{a}| \times |\vec{b}| \times \sin{\theta}\) ……where \(\theta\) is the angle between the vectors. The first result gives the cross product in component form. The second result gives the magnitude of the cross product.  An interesting application of cross product occurs in geometry. The magnitude of \(\vec{a} \times ...

Vector product of two non-parallel vectors is a vector perpendicular to both the vectors. If \(\vec{a}\) and \(\vec{b}\) are non-parallel vectors then \(\vec{a} \times \vec{b}\) is their cross product - a vector making 90 degree angle with both \(\vec{a}\) and \(\vec{b}\). It is also known as cross product. Another way to think of it is to imagine a plane “containing” the two non-parallel vectors. The vector product makes a 90 degree angle with the plane containing them, i.e. it is a vector normal to the plane .  Let \(\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}\) and,       \(\vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}\)   Then, \(\vec{a} \times \vec{b}\) is equal to the following determinant : \(\vec{a} \times \vec{b} = \Biggl|\matrix{\hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3}}\Biggl|\) The first row are the unit vectors \(\hat{i}, \hat{j}, \hat{k}...

There are two forms of equation of a plane : a vector form and a Cartesian form.  Further, there are two kinds of vector equation forms : a vector equation deduced using any normal(orthogonal or perpendicular) vector to the plane, and the other is called the parametric equation form which results from combination of vectors parallel to the plane. We have already looked at the former in one of my other posts. We will look at the latter now. The following analogy of a clock might help to get a basic idea in the beginning. In one complete revolution from 12 to 12, a minute hand(or any other hand of clock) points in every direction in the plane of the clock : N, NE, E, SE, etc… Or we can say that it points in every direction in the plane of the wall to which the clock is hung. To find a parametric equation of a plane in 3D space, we need a vector(similar to the minute-hand arrow), call it a variable vector, which can point in every possible direction in the given plane but mu...