12) The work W done by a constant force F as its point of application moves along the vector PQ is given by W = F· PQ. Find the work W done in moving a particle from point P(1,3) to Q(-3,5) when the magnitude and direction of the force are given by F = – 2i + 3j.

Q: find the angles between the vectors:x+y=1,2x+y-2z=2

A:

Q: 1 Suppose a line passes through the point (1, 1 and parallel to the 2' 3 plane x + y + z = 0. Then…

A: The direction vector of the normal to the plane is (1,1,1) Given that the line is parallel to the…

Q: Q1, Find Paametric equations for the line through the Point (0,1, 2) that is U =i 12}+ 3k and ū= 3i…

A:

Q: If vector u=2i+3j+4k is orthogonal to vector v=3i+2j-bk, then value of b is 4 1 2 3

A: Two vectors x , y  are orthogonal or perpendicular if x·y=0

Q: express vector as a product of its length and direction. 1/(3)^1/2 i + 1/(3)^1/2 j + 1/(3)^1/2 k

A: Let the given vector is: v=13i+13j+13k

Q: Find the work done by a force F = 4i – 6j + 9k that moves an object from the point (0, 10, 6) to the…

A:

Q: 1or Find. Ewo unit Vectors in X-y plane 30 deyree with the vectur that make angle cf V= 3i +4j ?

A:

Q: In Newton's theory of gravitation, the Earth's gravitational field is given by, -GM r3 where 7 = (x,…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: 3. Consider the vectors X and Y in R³ where X = -i+3j and Y = 4i-2j+ 2k. 1, Find the unit vector…

A: We can solve this as follows:

Q: Find a unit vector n that is normal to the surface 4z - 2x - 3y = 14 at P = (1,2,4) that points in…

A:

Q: What is the unit vector perpendicular aial yA n = i-j+ 3k, n2 = 2i + j + 3k to the vectors?

A: Given that:n1=i-j+3k,  n2=2i+j+3k

Q: The value of the chiral vector of SWNTs if the n=8 , m=6 and the first vector length equal n/2+1 and…

A: Given that m=6,n=8to find the chiral vector of SWNT.  As we know that, the chiral vector c can be…

Q: find a vector parametrization for the line with thegiven description.Passes through P = (1, 2, −8),…

A: Find a vector parametrization for the line with the given description. Passes through P = (1, 2,…

Q: 1) Find the MAGNITUDE and DIRECTION of the RESULTANT of ALL the forces at point A +Y -120lb APPLIED…

A: The resultant of vectors at a point is defined as the net effect of the forces acting at the given…

Q: 4. If the projections of vector a on x,y and z −axes are 2,1,2 units respectively, and the angle at…

A: To calculate the angle we use dot product of two vectors given by Taking two vectors as U&V…

Q: A force F = 2i +j - 3k is applied to a spacecraft with velocity vector v = 3i - j. Express F as a…

A:

Q: Q1, Find Parame tric equations for the line through the Point (0,1, 2) that is U =i +2} + 3k and ū=…

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: (a) Find the unit vector of 2q where q=(8 sin 0,-8 cos 0,0).

A:

Q: If a = 3i +j - 2k and b = 5i –j- 2k, then find %3D а. а xb b. vector resolute of vector a in…

A: We will find out the required values.

Q: find a vector parametrization for the line with thegiven description. Passes through P = (4, 0, 8),…

A: The parametric equation of a line in vector for is given by the expression r→=u→+tv→. Here u→ is the…

Q: 3. The line through the point (2, 2.4, 3.5) and parallel to the vector 3i+ j – k Answer + (2i+ 2.4j…

A:

Q: 6. Find the distance of the point (-1,1) to the line given in vector form (x, y) = (2,0) +t(4, 3).

A: Given that: The point is (-1, 1).The vector is:  (x, y)=(2, 0)+t(4, 3)

Q: Represent the line segment from P to Q by a vector-valued function and by a set of parametric…

A: Two vectors Px1,y1,z1 and Qx2,y2,z2 Line segment P to Q: 1) Parametric form is…

Q: Represent the line segment from P to Q by a vector-valued function and by a set of parametric…

A: we have to represent the line segment from P to Q by a vector valued function and by a set of…

Q: s the vector ở = (-1,–2,3) perpendicular to the plane 3x + 6y – 9z + 5 = 0? %3D

A:

Q: • We have 2x 2y Vf(x, y, z) = i+ +1)j+6k. x² + y²- 1 x² + y? – 1 | Thus Vf(1,1,0) = 27 + 37 + 6K.

A: We have to show the steps for the above equation.

Q: 3. A hiker begins at the origin and walks in a straight line in the direction of the vector (1,1)…

A: 1st we have to find points where hiker changes its direction and then draw figure of path  Distance…

Q: Find the angle 0 between the vectors. 5T u-(cos , sin S), v-(cos, sin 4) = 4 4 3

A:

Q: o v based at the origin.

A: Sketch the vector v = (2, −3) based at P = (1, 4) and the vector v0 equivalent to v based at the…

Q: Calculate the directional cosine cos⁡β of the vector .  Round your answer to the nearest hundredths.…

A:

Q: Find the Directional Derivabive of Pary,2)= x-yz'+ at the point CI, -2, 0) in the direcbion of…

A: A scalar function is a rule to associate a number to a particular point in space. The directional…

Q: Consider the line L with vector equation (x, y, z) = (2,3,0) + t(1, 4, –1). Determine the point on L…

A: Foot of perpendicular is the closest point on the given line, Let M is the foot of perpendicular…

Q: Evaluate The Work Done By The Force Vector, F= (6x^2+3y)I – 15yzj + 10x^2zk From (0,0,0) To (1,1,1)…

A:

Q: 3. Compute the angle between vector a = (1,2,3) and the positive r-axis.

A:

Q: Show that the vectors a = 3i+ 4j and b = 4i - 3j are perpendicular. A smooth sphere A of mass 2m…

A: Consider the given: a→=3i^+4j^b→=4i^-3j^ mass M of sphere A=2m velocity of sphere A of mass…

Q: Find a vector parametrization of line through P = (9, 8, 3) in the direction v = (4, -8, 6). %3D…

A:

Q: Find the work done in moving a particle from point P(- 5, 0, 2) to Q(3, - 1, 4) if the magnitude and…

A: To find the work done in moving a particle from point P-5,0,2 to Q3,-1,4 Also the magnitude and…

Q: Find the work done by a force F = 4i - 6j + 9k that moves an object from the point (0, 10, 6) to the…

A:

Q: Find the work done by a force F = 8i − 6j + 5k that moves an object from the point (0, 8, 6) to the…

A:

Q: 2. Can three vectors in the xy plane R2, have u v < 0, v w < 0, w u<0?

A: Given that, u, v and w are vectors in XY plane. The dot product of vectors is defined as, if u = a1…

Q: A force of 6 newtons is applied in the direction of the vector v = 2i + 2j - k. Express the force F…

A: Given vector, v=2i+2j-k

Q: A force F = 2i + j - 3k is applied to a spacecraft with velocity vector v = 3i - j . Express F as a…

A: We have given the force F = 2i + j - 3k with velocity vector v = 3i – j.

Q: What is the magnitude of a vector joining two points P(x1, y1, z1) and Q(x2, y2, z2)?

A: The two points  are : P(x1, y1, z1) and Q(x2, y2, z2). As the vector is directed from point P to Q ,…

Q: Let f(x, y) = 5y² ln(4x). Then Duf(1,4) in the direction of the vector (-2, –2) is

A:

Q: Find the work done by a force F = 4i - 8j + 9k that moves an object from the point (0, 8, 4) to the…

A:

Q: 1.23. Express the unit vector directed toward the point (x, Y, z) from an arbitrary point in the…

A: Express the unit vector directed toward the point (x, y, z) from an arbitrary point in the plane y =…

Q: Represent the line segment from P to Q by a vector-valued function and by a set of parametric…

A: Given information: The points are P(0, 0, 0), Q(5, 2, 2). Calculation: Find the vector valued…

Q: The surface of a hill can be given as h(x, y) = e-(z²+y²), If a ball is placed at the (2, y)…

A:

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.