Most Important Vectors And Equilibrium Quiz Test Online

The equation A · B = -AB implies that AB cosθ = -AB, so cosθ = -1. This occurs when the angle θ is 180°.

Reversing the order of multiplication in a vector product results in a vector with the same magnitude but opposite direction.

Using the right-hand rule: point your fingers North (A), then curl them towards West (B). Your thumb will point Up (out of the page).

Work is an example of a scalar product (W = F · d) and is therefore a scalar quantity. Force, torque, and velocity are vectors.

For parallel vectors, the angle θ is 0°. Since sin(0°) = 0, the magnitude of the vector product is zero, resulting in a null vector.

Angular momentum is a key physical example of the vector product, along with torque. The other options are scalar quantities.

Using |A x B| = AB sinθ. So, |A x B| = 2 × 3 × sin(30°) = 6 × 0.5 = 3.

The scalar product of two vectors obeys the commutative law, meaning the order of the vectors does not affect the result.

The magnitude of the cross product, ABsinθ, gives the area of the plane (parallelogram) determined by the two vectors A and B.

The component of vector B that is parallel to A is its projection onto A. This is given by B cosθ.

The direction of the vector product is determined by the right-hand rule, where rotating the fingers from the first vector to the second gives the direction of the thumb as the resultant.

The cross product of i and j is a vector perpendicular to both, which is k. The magnitude is |i||j|sin(90°) = 1. The direction is determined by the right-hand rule.

The magnitude is |A x B| = AB sinθ. The value of sinθ is maximum (equal to 1) when θ = 90°.

If vectors are in the same direction, θ = 0° and cos(0°) = 1. So, A · B = AB(1) = AB.

Torque is explicitly defined as the vector product of the position vector r and force F. Work is a scalar product.

Because a dot is placed between the two vectors, the scalar product is also called the dot product.

A cross (x) is placed between the two vectors to represent this multiplication, hence it is also called the cross product.

A is along the x-axis and B is along the y-axis, so they are perpendicular. The dot product of perpendicular vectors is always zero.

The angle between a vector and itself is 0°. Since sin(0°) = 0, the magnitude of the cross product is zero, resulting in a null vector.

When a vector is multiplied by a vector and the resultant is a vector quantity, the multiplication is called a vector product.

Vectors can be added geometrically by drawing them to scale and placing them head to tail. The resultant vector is drawn from the tail of the first vector to the head of the last one.

When a body is at rest under the action of several forces, it is in static equilibrium. A book on a table is a classic example.

The quantity d = r sinθ, which is the perpendicular distance from the rotation axis to the line of action of the force F, is called the moment arm or lever arm.

A couple is defined as two parallel forces that have the same magnitude but opposite directions and are separated by a perpendicular distance. Its only effect is to produce rotation.

The first condition of equilibrium is satisfied when the vector sum of all forces acting on the body is zero (ΣF = 0). This ensures translational equilibrium.

When a body moves with uniform linear velocity, it is said to be in dynamic translational equilibrium. A paratrooper falling at a constant (terminal) velocity fits this description.

Statics is the branch of mechanics that deals with the study of objects in equilibrium.

Torque, or the moment of force, is defined as the turning effect produced in a body about a fixed point due to an applied force.

The document states that by convention, anti-clockwise torques are taken as positive and clockwise torques as negative.

Graphically, a vector is shown by an arrow where the length of the arrow gives the magnitude (under a certain scale) and the arrow head points in its direction.

For a Cartesian coordinate system, the unit vector along the x-axis is î, along the y-axis is ĵ, and along the z-axis is k̂.

The magnitude of torque is τ = rF sinθ. The sine function is zero at both 0° and 180°, which means the torque will be at its minimum (zero) when the force is applied parallel or anti-parallel to the position vector.

The document defines a null vector as a vector having zero magnitude and an arbitrary direction. It can be obtained, for example, by subtracting a vector from itself.

The document defines a vector as a mathematical quantity having both magnitude and direction. Examples include weight and velocity.

Since F_net = m*a_net, if the net force is zero, the net translational acceleration must also be zero. This guarantees translational equilibrium only.

An equilibrant force is a single force that can balance two or more concurrent forces. It is equal in magnitude but opposite in direction to the resultant force.

The second condition of equilibrium is satisfied when the vector sum of all torques acting on the body is zero (Στ = 0). This ensures rotational equilibrium.

The unit vector is obtained by dividing the vector by its own magnitude, as shown by the formula  = A/|A|.

The starting point of a vector is called the tail, and the ending point is called the head.

Torque is maximized when the angle between the position vector (from the hinge to the handle) and the applied force is 90°, as sin(90°) = 1.

Vector subtraction is defined as the addition of the negative of the vector. So, A - B is the same as A + (-B).

When two or more forces act on a body and their lines of action pass through a common point, the forces are said to be concurrent.

Torque is defined as the vector product of the position vector r and the applied force F. The order matters due to the anti-commutative nature of the cross product.

A unit vector is defined as a dimensionless vector with a magnitude of 1 used to represent the direction of a vector.

For an object to be in complete equilibrium, it must be in both translational and rotational equilibrium. Therefore, both the first (ΣF=0) and second (Στ=0) conditions must be satisfied.