Dot Product And Cross - Free MCQ Practice Test with solutions, GATE Civil

The component of the force in the u axis, it is equal to 100cos(45˚). We need to consider the triangle and then accordingly apply the trigonometry. This is one of the way of resolving the components.

The dot product of the two vectors is always the product of the magnitudes of the two forces and the cosine of the angle between them. We need to consider the triangle and then accordingly apply the trigonometry. This is one of the way of resolving the components.

For three vectors A, B and D the various laws are. Communitive law: A.B =B.A. While distributive law is A.(B+D) = (A.B) + (A.D). And multiplication law is a(A.B) = A.(aB).

For three vectors A, B and D the various laws are. Communitive law: A.B =B.A. While distributive law is A.(B+D) = (A.B) + (A.D). And multiplication law is a(A.B) = A.(aB).

For three vectors A, B and D the various laws are. Communitive law: A.B =B.A. While distributive law is A.(B+D) = (A.B) + (A.D). And multiplication law is a(A.B) = A.(aB).

The multiplication of x, y and z components with their respective same component give a scalar, equal to 1, i.e. i.i = 1 and j.j = 1, while jxj =0. This is the basic principle of the vector algebra which needs to apply wherever needed.

Force component in the direction parallel to the AB is given by unit vector 0.286i + 0.857j + 0.429k. Now (300j).(0.286i + 0.857j + 0.429k) = 257.1N. Just try to resolve the force into it’s particular components.

As the dot product of only the same Cartesian component is unity, i.e. i.i = 1 and j.j =1, rest all remaining dot product will give 0(i.j = 0 and j.k = 0). Cross product of the same plane vectors always give zero. And dot product of the same plane vector gives a scalar quantity.

{(i.i) + (-i.j) + (-k.k) + (k.i)} = 0. As (1 + 0 + 0 – 1). Cross product of the same plane vectors always give zero. And dot product of the same plane vector gives a scalar quantity.

Cosine inverse function is defined only between 0˚ to 180˚. It cannot be defined under any of the given range, because this is the principle range of the inverse cosine function.

Here we are trying to do the cross product of the two vectors in the same plane. Which will give us zero. While dot product of the same plane vector will give a scalar quantity, not zero.

the cross-product needs to be taken in the proper sequence.

If not taken, then the answer is just the opposite of the true answer. That's why the answer is not the Force and Radius vectors but the Radius and Force vectors.

Because the moment has its direction, as many of the cross products have, precaution needs to be taken.

Fab= -(3/5)j + (4/5)k ; F= (4/5)i – (3/5)j ; θ = cos-1(Fab.F) = 69˚. This is the application of the triangle over the figure. Try to resolve the components of the given force. It will be easy.

Force’s component in the direction perpendicular to AB is -73.5i + 80j – 110k N. This is the application of the triangle over the figure. Try to resolve the components of the given force. It will be easy.

The component of the force in the v axis, it is equal to 100cos(15˚). This is the application of the triangle over the figure. Try to resolve the components of the given force. It will be easy.

The dot product of two vectors having the angle between them equal to 45° will have the product of the vector’s magnitude. As the vectors are of unit magnitude, their product will be unity. Thus the magnitude factor would be cosine function at 45°.

The cross product of two vectors gives a vector which is perpendicular to both of the vectors. And the mathematic equation for the same is given by: |A||B|sinØ. And the dot product of the same by any of the other two vectors will give the answer zero, as perpendicular.

This statement is wrong. It is not possible, unless we apply a negative sign to the RHS of the equation. That is PxQ = -(QxP). It is because, if you curl your wrist from one vector towards another vector, the thumb projected will give the direction of the cross product. Thus if you reverse the direction, negative sign is necessary.

The distributive law works just like the simple multiplication of the constant before the brackets. That is in the equation Px(Q+S) = (PxQ) + (PxS), P is crossed by Q and S both. This is simple as, if we first add the two vectors and then do the cross product or we first do the cross product and the do the sum.

The associative law is defined in the cross product of three vectors. This property is though valid for the dot product. But for the cross product, it is not true. It is because, in the dot product the final result is the scalar quantity, but in cross product it is the direction too. Thus the answer.

As the mathematic equation for the cross product is having a cosine function in it, in which the angle used in the function is the angle between the vectors. Thus the cross product will be zero if the angle between them is 90.

As asked, the one which is not correct is the third one. The product is containing the cosine function, and the angle which is going to be inserted in the function is the angle between the vectors. Thus if the angle is 90, then the cross will be zero.

The concurrent forces are the which are somewhere touching the axis of rotation. If any of the force is touching that axis, that force is not considered, or is insufficient to cause a rotation. If a force is concurrent then the perpendicular distance of the force from the line of axis is zero, thus no rotation. As we know rotation is caused by moment.

The moment of the force about a axis or a point gives the measure of the tendency of the force. It is the cause of the body’s rotation, about that point or at that axis. Thus, the tendency of a force to rotate the body is called the moment of the force.

The cross product needs to take in the proper sequence. If not taken then the answer is just the opposite of the true answer. That’s why, the answer is not the Force and Radius vectors, but the Radius and Force vectors. Because the moment has its direction, as many of the cross products have, and thus precaution needs to be taken.

For many vectors acting on the body, Varigon’s Theorem is defined as M = R x F (F = F₁ + F2 + F₃ + … vectorially adding all the forces). This is just the giving the cross product of individual force and radius vectors and thus the sum is the total moment produced.

It is the scalar expression. It is the dot product of the vector to the vector, which is the result of the cross vector product of two vectors. And if two vectors are held together in the equation by a dot “.”, then the answer which you get is the scalar quantity.

The calculation for finding the angle with the z-axis is based on simple geometry:

•  the angle between the vector and a perpendicular axis can be found by subtracting the given angle from 90°.
• The angle is 30° because it is derived from 90° minus 60°.

the answer is 30°.

Thus answer is option c)

The dot product of any two vectors is having the cosine function in it. It is being multiplied by product of the magnitudes of the vectors. So if the angle is 90°, then the value of cosine function would be zero, thus 0. And the value of the sine angle is 1, thus the result.

The length of each vector and the angle between them represent their magnitude and direction of the action line, respectively. The head/tip of the arrow indicates the sense of direction of the vector.

option A is correct