Vector Assignment 2

Problem 1.  Compute the magnitude of the vector \vec{v} = \langle \, 1, 2, 3, 4, 5 \, \rangle and compute the magnitude of the vector \vec{v} - 2\vec{w} where \vec{w} = \langle \, 1/2, 3, 0, 2, -1 \, \rangle.

Problem 2.  (a) Draw a picture of the vector \vec{v} = \langle\, 1, 2 \, \rangle in standard position in \mathbb{R}^2.

(b) Find all numbers c \in \mathbb{R} so that \vec{v} (from part (a) above) is perpendicular to the vector \vec{w} = \langle \, 1, c \rangle.  Once you have found all possible values for c, pick one and draw the vector \vec{w} in the same picture you from part (a) where you drew the vector \vec{v}.

Problem 3. (a)   Draw a picture of the set of points S \subset \mathbb{R}^2 defined by

\displaystyle S = \left \{ (x,y) : x^2-2x + y^2 = 8 \right\}.

(b) Draw a picture of the set of points S \subset \mathbb{R}^3 defined by

\displaystyle S = \left \{ (x, y, z) : x^2 - 4x  + y^2 - 2y  + z^2 = -4 \right\}.

Problem 4.  Complete the definition of the dot product between two vectors \vec{v}, \vec{w} \in \mathbb{R}^n where \vec{v} = \langle \, v_1, v_2, \cdots, v_n \, \rangle and \vec{w} = \langle \, w_1, w_2, \cdots, w_n \, \rangle:

\displaystyle \vec{v}\cdot \vec{w} =

Problem 5.  Exercise 1 from Section 9.3 of our textbook.

Problem 6.  Exercise 3 from Section 9.3 of our textbook.

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