Question

What is the shear transformation for the vector 3i^+2j^3 \hat{i} + 2 \hat{j}3i^+2j^?

Answer 1

Answer:

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Answer 2

Given vector = $3i+2j$

$A transformation that slants the shape of an object is called the shear transformation$

$There are two kinds of shear transformation.$

1. $horizontal shear(shear parallel to x axis)$  

2. $vertical shear(shear parallel to y axis.)$

 $For finding the shear transformation let assume shear factor is x$

$1.horizontal shear(shear parallel to x axis)→ shearing will displace point in right direction,if m is (+),and will move point in left if m is$$shearing of (3,2) in x-axis direction direction by value m=(3+2m, 2) as the point (3,2) moves to a new coordinate(3+2m,2).$

$suppose earlier the angle between the point and origin was \alpha \\ after shearing it becomes \beta , so shearing angle is cot\beta which is cotangent to \alpha$

$2. shearing vertically(along vertical axis i.e parallel to y-axis)→shearing the point (3,2) by an amount along parallel to y-axis in the right direction will take it to (3,2+3 m).shearing will displace point in the right direction. if m is a positive direction, and will move the point in left if m is of the angle made by point with the origin is the shearing angle.$

$\therefore$ Shearing transfomation is $(3+2m,2) and (3,2+3m)$ in the repective direction.