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3D Vectors (R^{3})

The properties and the operations of 3D vectors are explained.

What is a Vector?
A
In the above, the vector is defined using an Equivalent Vectors Vectors with equal magnitudes and same direction are equivalent vectors. Sum of two Vectors Given two vectors \( \vec{v_1} \) and \( \vec{v_2} \), their sum is a vector obtained by first positioning vector \( \vec{v_2} \) such that its initial point coincide with the terminal point of \( \vec{v_1} \) and the sum \( \vec{v_1} + \vec{v_2} \) is the vector whose initial point is the initial point of \( \vec{v_1} \) and its terminal point is the terminal point of \( \vec{v_2} \). Note that \( \vec{v_1} + \vec{v_2} = \vec{v_2} + \vec{v_1} \). Also the sum of two vectors coincide with the diagonal of the parallelogram determined by \( \vec{v_1} \) and \( \vec{v_2} \).
Difference of two Vectors Given two vectors \( \vec{v_1} \) and \( \vec{v_2} \), the difference \( \vec{v_2} - \vec{v_1} \) may be defined as a sum \( \vec{v_2} + (- \vec{v_1}) \) and represented geometrically as shown below.
Multiplication of a Vector by a Scalar A vector \( \vec{v_1} \) multiplied by a scalar \( k \) is defined as a vector \( k\vec{v_1} \) parallel to \( \vec{v_1} \) and whose direction is the same as that of \( \vec{v_1} \) if k > 0 and opposite if k <0 . The magnitude (length) of \( k\vec{v_1} \) is \( | k | \) times the magnitude of \( \vec{v_1} \). Figure below shows vectors \( \vec{v_1} \), \( 2\vec{v_1} \) and \( -3\vec{v_1} \).
Vectors in a 3D Rectangular Coordinate System
A unit vector is a vector with magnitude equal to 1. Below is a shown a 3D rectangular coordinate system with unit vectors \(\vec{i} \), \(\vec{j} \) and \(\vec{k} \) in the positive direction of the x, y and z axes respectively. Vectors \(\vec{i} \), \(\vec{j} \) and \(\vec{k} \) may be defined by their
Components of a Vector The
The components of a vector \(\vec{v} \) defined by its initial point \( A = (x_1 , y_1 ,z_1)\) and its terminal point \( B = (x_2 , y_2 ,z_2) \) are given by
Use Components of a Vector to Calculate its Magnitude and Unit vector in the Same Direction Given vector \( \vec{v} = < a,b,c> \), its magnitude (or length) is given by\( ||\vec{v}|| = \sqrt{a^2+b^2+c^2} \). The unit \( \vec{u} \) vector, defined as a vector of magnitude equal to 1, in the same direction as \( \vec{v} \) is given by
\( u = \dfrac{1}{||\vec{v}||} \vec{v} \)
Use Components of Vectors to Calculate the Sum, Difference, and Scalar multiplication of Vectors Given vectors \( \vec{v_1} = < a_1,b_1,c_1> \) and \( \vec{v_2} = < a_2,b_2,c_2> \), the sum \( \vec{v_1} + \vec{v_2}\) , the difference \( \vec{v_1} - \vec{v_2}\) and scalar multiplication \( k \vec{v_1} \), k a real number, are given by\( \vec{v_1} + \vec{v_2} = < a_1+a_2,b_1+b_2,c_1+c_2> \) \( \vec{v_1} - \vec{v_2} = < a_1 - a_2,b_1 - b_2,c_1 - c_2> \) \( k \vec{v_1} = < k a_1,k b_1,k c_1> \)
Detailed Solutions and explanations to the questions below are included. 1) Find the components of the vectors \( \vec{AB} \) and \( \vec{BA}\) where A and B are points given by their coordinates A(2,6,7) and B(0,-3,1) and show that \( \vec{AB} = -1 \vec{BA}\). 2) Given vectors \(\vec{v_1} = <0,-3,2>\) and \( \vec{v_2} = <-3,4,5> \), find: a) \( \vec{v_1} + \vec{v_2} \) b) \( \vec{v_1} - \vec{v_2} \) c) \( -3\vec{v_1} \) d) \( -2\vec{v_1} + 3\vec{v_2} \) e) \( k \)such that \( ||\vec{v_1} + k\vec{v_2}|| = \sqrt{67} \). 3) Given vector \(\vec{v} = <0,-3,2>\), find the unit vector in the same direction as \(\vec{v} \) and check that its magnitude is equal to 1. 4) Given the points A(2,6,7), B(0,-3,1) and C(0,3,4), find the components of the vectors \( \vec{AB} \), \( \vec{AC}\) and \( \vec{BC}\) and show that \( \vec{AB} + \vec{BC} = \vec{AC}\). 5) Given the points A(-1,2,1), B(2,4,2) and C(5,6,3), find the components of the vectors \( \vec{AB} \), \( \vec{BC}\) and \( \vec{AC}\) and determine which of these vectors are equivalent and which are parallel. 6) Given vectors \(\vec{v_1} = <-4,0,2>\) and \( \vec{v_2} = <-1,-4,2> \), find vector \( \vec{v} \) such that \(\vec{v_1} - 2 \vec{v} = 3 \vec{v} - 3 \vec{v_2} \) 7) Find a vector in the same direction as vector \( \vec{v} = <-4,2,2> \) but with twice the length of \( \vec{v} \). 8) Find a vector in the opposite direction of vector \( \vec{v} = <-1,2,2> \) but with a length of 5 units. 9) Given vector \( \vec{v} = <-1,2,2> \), find a real number \( k \) such that \( ||k \vec{v} || = 1/5 \). 10) Find \( b \) and \( c \) such that vectors \(\vec{v_1} = <-4,6,2>\) and \( \vec{v_2} = <2,b,c> \) are parallel. 11) Are the three points A(2,6,7), B(1,4,5) and C(0,2,3) collinear? 12) A cube of side 2 units is shown below. a) Find the components of the vectors \( \vec{AB} \), \( \vec{EF} \), \( \vec{DC} \), \( \vec{HG} \), \( \vec{AC} \) and \( \vec{AG} \). b) Which of the vectors in part a) are equivalent? c) Prove algebraically that \( \vec{AB} + \vec{BF} + \vec{FG} = \vec{AC} + \vec{CG} \). d) Find \( || \vec{AG} || \). e) Find the unit vector in the same direction as vector \( \vec{AG} \).
Detailed Solutions and explanations to these questions. |

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