Questions on torque and cross products

[prime-factor]

Homework Statement

Hello. I have three questions on vector product that I am having problems with.
I have attached the images for these questions.

Homework Equations

Vector Product:

a x b = lal lbl sin (theta) n

or i, j, k method

Area of a triangle:

(1/2) l a x b l

Torque = l r x F l, where r = vector from force, F = force vector

The Attempt at a Solution

I am confused about these questions. I would really appreciate your help in understanding how to work them out.

Thanks in advance.

 

[mighty2000]

Dear student,

I am happy to help you with your questions on vector product. I understand that it can be a challenging concept to grasp, but with practice and understanding of the underlying principles, you will be able to solve these questions easily.

Firstly, let's review the equation for vector product:

a x b = lal lbl sin (theta) n

This equation represents the vector product, also known as the cross product, of two vectors a and b. The result of this operation is a vector that is perpendicular to both a and b, and its magnitude is equal to the product of the magnitudes of a and b multiplied by the sine of the angle between them. The direction of the resulting vector is determined by the right-hand rule, where you point your fingers in the direction of a and then curl them towards b. Your thumb will then point in the direction of the resulting vector.

Now, let's apply this to the first question. We are given two vectors a and b, and we are asked to find their cross product. Remember that the resulting vector is perpendicular to both a and b, so it will be in the direction that is perpendicular to the plane containing these two vectors. To find the direction, we can use the right-hand rule as described above. Once we have the direction, we can use the equation to calculate the magnitude of the resulting vector.

Moving on to the second question, we are given three vectors a, b, and c, and we are asked to find the area of the triangle formed by these vectors. The formula for the area of a triangle using vector product is:

(1/2) l a x b l = (1/2) l c l l sin (theta) l

where c is the third side of the triangle and theta is the angle between a and b. This formula comes from the fact that the area of a triangle is equal to half the product of its base and height, and in this case, the base is the magnitude of c and the height is the sine of the angle between a and b.

Finally, in the third question, we are asked to find the torque exerted by a force on a lever. The formula for torque is:

Torque = l r x F l

where r is the vector from the point of rotation to the point of application of the force, and F is the force vector. To find the torque, we need to find the cross product of r