1.1.1 Show how to find A and B, given A+B and A −B.
1.1.2 The vector A whose magnitude is 1.732 units makes equal angles with the coordinate axes. Find Ax,Ay , and Az. 1.1.3 Calculate the components of a unit vector that lies in the xy-plane and makes equal angles with the positive directions of the x- and y-axes. 1.1.4 The velocity of sailboat A relative to sailboat B, vrel, is defined by the equation vrel = vA − vB, where vA is the velocity of A and vB is the velocity of B. Determine the velocity of A relative to B if vA = 30 km/hr east

vB = 40 km/hr north.
ANS. vrel = 50 km/hr, 53.1◦ south of east.
1.1.5 A sailboat sails for 1 hr at 4 km/hr (relative to the water) on a steady compass heading of 40◦ east of north. The sailboat is simultaneously carried along by a current. At the end of the hour the boat is 6.12 km from its starting point. The line from its starting point to its location lies 60◦ east of north. Find the x (easterly) and y (northerly) components of the water’s velocity. ANS. veast = 2.73 km/hr, vnorth ≈ 0 km/hr.

1.1.6 A vector equation can be reduced to the form A = B. From this show that the one vector equation is equivalent to three scalar equations. Assuming the validity of Newton’s second law, F = ma, as a vector equation, this means that ax depends only on Fx and is independent of Fy and Fz. 1.1.7 The vertices A,B, and C of a triangle are given by the points (−1, 0, 2), (0, 1, 0), and (1,−1, 0), respectively. Find point D so that the figure ABCD forms a plane parallelogram. ANS. (0,−2, 2) or (2, 0,−2). 1.1.8 A triangle is defined by the vertices of three vectors A,B and C that extend from the origin. In terms of A,B, and C show that the vector sum of the successive sides of the triangle (AB +BC +CA) is zero, where the side AB is from A to B, etc. 1.1.9 A sphere of radius a is centered at a point r1. (a) Write out the algebraic equation for the sphere.

(b) Write out a vector equation for the sphere.
ANS. (a) (x −x1)2 +(y −y1)2 + (z...

...3r21. ABCD is a rectangle and O is the midpoint of [AB].
Express each of the following vectors in terms of and
(a)
(b)
(c)
(Total 4 marks)
2. The vectors , are unit vectors along the x-axis and y-axis respectively.
The vectors = – + and = 3 + 5 are given.
(a) Find + 2 in terms of and .
A vector has the same direction as + 2 , and has a magnitude of 26.
(b) Find in terms of and .
(Total 4 marks)
3. The circle shown has centre O and radius 6. is the vector , is the vector and is the vector .
(a) Verify that A, B and C lie on the circle.
(3)
(b) Find the vector .
(2)
(c) Using an appropriate scalar product, or otherwise, find the cosine of angle .
(3)
(d) Find the area of triangle ABC, giving your answer in the form a , where a ∈ .
(4)
(Total 12 marks)
4. The quadrilateral OABC has vertices with coordinates O(0, 0), A(5, 1), B(10, 5) and C(2, 7).
(a) Find the vectors and .
(b) Find the angle between the diagonals of the quadrilateral OABC.
(Total 4 marks)
5. Find a vector equation of the line passing through (–1, 4) and (3, –1). Give your answer in the form r = p + td, where t ∈ R
(Total 4 marks)
6. In this question, the vector km represents a displacement due east, and the vector km a...

...MIchael Driesen
Mrs. Rozell
Math 10H
17 December 2011
Vectors
Math is everywhere. No matter which way you look at it, it’s there. It is especially present in science. Most people don’t notice it, they have to look closer to find out what it is really made of. A component in math that is very prominent in science is the vector. What is a vector? A vector is a geometric object that has both a magnitude and a direction. A good example of a vector is wind. 30 MPH north. It has both magnitude,(in this case speed) and direction. Vectors have specific properties that make them very useful in real life applications. Through the use of these special objects, many advancements in the fields of math and science are available.
Representations
Vectors can sometimes be hidden behind basic objects. They are usually represented with an arrow on top of its starting point and terminal point, as shown here: The most common form of vector is the bound vector. All that means is that the starting point of the vector is the origin, or (0,0). The bound vector goes from the origin to it’s terminal point, which in this case can be (3,4). An easy way to write this
is A = (3,4), where A is the vector. On a graph, it looks like this:
That applies for two-dimensional vectors. Three dimensional...

...ECE 352 VECTORANALYSIS
DEL OPERATOR
GROUP 3 Andaya, Rizalyn Ramos, Maria Issa P.
∇
Del is a symbol used in mathematics, in particular, in vector calculus, as a vector differential operator, usually represented by the nabla symbol ∇. Del may denote the gradient (locally steepest slope), the divergence of a vector field, or the curl (rotation) of a vector field. The symbol ∇ can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product of scalars, dot product, and cross product, respectively, of the del "operator" with the field.
A. GRADIENT If a scalar function , , is continuously differentiable with respect to its variables x, y, z through the region, then the gradient of , written grad , is defined as the vector grad = Using the vector differential operator ∇, ∇= We can write the gradient of as
+
+
+
+
grad = ∇ = ( = Example: If =
+ +
+ +
)
, determine the grad at the point P(1,3,2).
By definition: grad = ∇ =
+
+
All we have to do then is to find the partial derivatives at x=1, y=3, z=2 and insert their values.
Since = = =
= = = = 2(1) (3) (2)3 + (3)2(2)2 (1)2(2)3 + 2(1) (3) (2)2 3(1) (3) (2)2 + 2(1) (3)2(2) = 84 = 32 = 72
grad
= ∇ = 84 i + 32 j +72 k
B. DIVERGENCE If a...

...VectorAnalysis
Definition
A vector in n dimension is any set of n-components that
transforms in the same manner as a displacement when you
change coordinates
Displacement is the model for the behavior of all vectors
Roughly speaking: A vector is a quantity with both direction as well as
magnitude.
On the contrary, a scalar has no direction and remains unchanged when
one changes the coordinates.
Notation: Bold face A, in handwriting A . The magnitude of the vector is
denoted by A A A
Example:
Scalars: mass, charge, density, temperature
Vectors: velocity, acceleration, force, momentum
Vector Algebra
Vector Operations
(a) Addition of two vectors
Parallelogram law: To find A+B, place the tail of B at the head of A and
draw the vector from the tail of A to the head of B
B
A
A+B
From the definition, the addition of vectors is
(i) Commutative
A+B=B+A
(ii) Associative
(A+B)+C=A+(B+C)
(b) Negative of a vector
The negative of a vector is defined as the vector with the same magnitude
but opposite direction
A
-A
(c) Multiplication by a scalar
Multiplication by a positive real number a multiplies the magnitude by a
times while leaving the direction unchanged.
A
A
aA
aA
Multiplication by 0 gives the null...

...Research paper about vector organisms such as ticks and mosquitoes and ways to remove them.
Vector Organisms and the Diseases They Carry
Vector organisms are parasites that transmit various diseases. Two vector organisms are ticks and mosquitoes. These two vector organisms transmit many well-known diseases. Through most of this essay, I will be explaining these two different organisms and the diseases that they carry.
First, I will be explaining deer ticks. Ticks are small spider-like organisms that attach themselves to a donor body. Male ticks are black. Females have a red abdomen and some black near the head. Adult ticks usually feed on deer but some times feed on humans accidentally. Deer ticks go through three life stages. The first stage is the larva. The larvae are small and tan colored. They usually feed on things like chipmunks, mice, and other small animals. Nymphs are a little bit larger than the larva. They are beige with a dark head. Nymphs usually feed on larger animals such as birds and raccoons. The adults vary in color depending if they are males or females. They usually feed on cats, dogs, and humans. Deer ticks are usually found in grassy woodland areas or wide open plains. Deer ticks are more common in the eastern U.S. than in the western U.S.
Mosquitoes are small insects that have a very thin body. Mosquitoes can be found anywhere except for Antarctica. There are about 170...

...screening test, chagas immunoblot test and fungal pathogen molecular test. Immunetics develops and markets various FDA-cleared products, including a test for Lyme disease. The company offers customers support services. It markets its products worldwide. Immunetics is headquartered in Boston, Massachusetts, the US.
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...object is a widely understood concept. The magnitude of the force acting on the object is a simple number, referred to as a scalar. In order to fully understand the effect of this force on the object, the direction in which the force is being applied must be specified. This combination of a scalar force and a mathematical description of the direction in which the force is acting is referred to as a vector. It is not enough to know that a thousand pounds of force is being exerted on a person. A thousand pounds of force can crush the person into the earth or send the person flying into the air, depending on the vector. Vector mathematics can be used to calculate the resultant effect when more than 1 vector is acting on an object. Vector mathematics also can be used to take the recorded effect on an object and back calculate what vectors must have combined to cause that effect. Use these tips to learn how to work out vectors in physics.
Steps
Orient the Vectors1
-------------------------------------------------
1. Determine a common frame of reference. The directions of the vectors must relate to each other in a common frame of reference in order to add or subtract them. The accepted way is to define space in axes: x, y, and z. These axes are perpendicular to each other.
* Declare the x axis. For example, use a standing person and state that the x axis...