Area of parallelogram in vector form

The area of parallelogram is the region covered by a parallelogram in a two-dimensional plane. A parallelogram is a two-dimensional figure with four sides and the area of parallelogram is the space enclosed within its four sides and is equal to the product of its length and height. A parallelogram is a special quadrilateral case, where opposite sides are equal and parallel. As it is a two-dimensional figure, it has an area and perimeter. The sum of the interior angles in a parallelogram is 360 degrees. More about the area of parallelogram along with its formula, derivations, and more solved problems are covered in this article in detail. Table of Contents • • • • • • • • What is the Area of Parallelogram? The area of parallelogram is the region it bounds in a given two-dimension space. A parallelogram is a special type of quadrilateral with a pair of parallel opposite sides. The opposite sides of a parallelogram are of equal length and the opposite angles are of equal measures. Formula for the Area of Parallelogram To find the area of the parallelogram, multiply its base by its height. The base and the height of the parallelogram are perpendicular to each other and the lateral side of the parallelogram is not perpendicular to the base. Therefore, a dotted line is drawn to represent the height. Area of parallelogram = b × h square units Where ‘b’ is the base of the parallelogram, ‘h’ is the height of the parallelogram. How to Calculate the Area of Parallelogram? The area of...

That is a good line of thought. This is what i think....Having given it a thought. A vector has magnitude and direction. It has a zero determinant. Given a linear function above the x-axis(for simplity), the integral of the function is the area under the graph. This linear function can also be thought of as line kV1(scaler multiples of some vector v1) through the origin or Vo + kV1 where the scaler multiples start off from a vector Vo. This line can be seen as a matrix-vector product T = AX taking an input Rn and mapping it to R1. Meaning it takes a vector in Rn and squishes it to a line. Now finding the determinant of A(the transformation matrix) is 0. det(A). That is, the determinant of the transformation matrix is 0 and the determinant of the line (if viewed as a long vector) is also zero. Nonetheless, the area below the line may not be zero but the determinant will always be zero. The case gets 🤢 if the function is not linear. Probably some crazy curve. The integral is still the area under the curve. But what is the determinant? This is not a linear transformation mind you, and not a square matrix(n,n)... because it maps from some Rn to R1. And so no determinant. This transformation matrix A takes a vector in Rn and maps it onto a point. HA! So T = AX where A is 1 by n matrix and X is n by 1 vector in Rn. T = A(1,n) * X(n, 1) Happy Learning_HL I've got a 2 by 2 matrix here, and let's just say its entries are a, b, c, and d. And it's composed of two column vectors. We'v...

4 Differentiation of Functions of Several Variables • Introduction • 4.1 Functions of Several Variables • 4.2 Limits and Continuity • 4.3 Partial Derivatives • 4.4 Tangent Planes and Linear Approximations • 4.5 The Chain Rule • 4.6 Directional Derivatives and the Gradient • 4.7 Maxima/Minima Problems • 4.8 Lagrange Multipliers • 5 Multiple Integration • Introduction • 5.1 Double Integrals over Rectangular Regions • 5.2 Double Integrals over General Regions • 5.3 Double Integrals in Polar Coordinates • 5.4 Triple Integrals • 5.5 Triple Integrals in Cylindrical and Spherical Coordinates • 5.6 Calculating Centers of Mass and Moments of Inertia • 5.7 Change of Variables in Multiple Integrals • Learning Objectives • 2.4.1 Calculate the cross product of two given vectors. • 2.4.2 Use determinants to calculate a cross product. • 2.4.3 Find a vector orthogonal to two given vectors. • 2.4.4 Determine areas and volumes by using the cross product. • 2.4.5 Calculate the torque of a given force and position vector. Imagine a mechanic turning a wrench to tighten a bolt. The mechanic applies a force at the end of the wrench. This creates rotation, or torque, which tightens the bolt. We can use vectors to represent the force applied by the mechanic, and the distance (radius) from the bolt to the end of the wrench. Then, we can represent torque by a vector oriented along the axis of rotation. Note that the torque vector is orthogonal to both the force vector and the radius vector. In this ...