5.1 2D Vectors

Let [math]A(3,7), B(-1,6), C(1,8)[/math], and [math]D(0,-1)[/math] be 4 points.

(a) If [math]\|\overrightarrow{CA}\|=\sqrt{a}[/math] where [math]a[/math] is an integer, what is [math]a[/math]?

(b) Find [math]\overrightarrow{AB}-2\overrightarrow{CD}[/math].

(c) Find [math]a[/math] and [math]b[/math] such that [math]a\overrightarrow{AB}+b\overrightarrow{DC}=\overrightarrow{0}[/math].

Consider four points [math]A, B, C[/math], and [math]D[/math] and these points can form a square. How many distinct vectors can be determined?

Let [math]\overrightarrow{a}=5\textbf{i}+3\textbf{j}[/math] and [math]\overrightarrow{b}=-6\textbf{i}+5\textbf{j}[/math].

(a) Find [math]5(\overrightarrow{a}+4\overrightarrow{b})[/math].

(b) Which of the following vectors is the unit vector of [math]\overrightarrow{a}[/math]?
(A) [math]-5\textbf{i}-3\textbf{j}[/math]
(B) [math]\dfrac{5}{\sqrt{8}}\textbf{i}+\dfrac{3}{\sqrt{8}}\textbf{j}[/math]
(C) [math]-\dfrac{5}{\sqrt{34}}\textbf{i}-\dfrac{3}{\sqrt{34}}\textbf{j}[/math]
(D) [math]\dfrac{3}{\sqrt{34}}\textbf{i}+\dfrac{5}{\sqrt{34}}\textbf{j}[/math]

Consider a point [math]A(2,-5)[/math] and a vector [math]\overrightarrow{u}=4\textbf{i}-3\textbf{j}[/math]. Let [math]B[/math] be a second point so that [math]\overrightarrow{AB}[/math] is parallel to [math]\overrightarrow{u}[/math] and [math]\|\overrightarrow{AB}\|=7[/math]. If [math]B[/math] is not in the fourth quadrant, find the coordinates of [math]B[/math]?

Let [math]A(3,0), B(1,-5)[/math] and [math]C(-7,-6)[/math] be three points. Find the coordinates of point [math]D[/math] such that [math]ABCD[/math] can form a parallelogram.

(Calculator) Three forces [math]F_1, F_2[/math], [math]F_3[/math] are applied on an object. The figure shows [math]F_1[/math] and [math]F_2[/math] and the object. The magnitudes of [math]F_1[/math] and [math]F_2[/math] are 3 and 5 respectively. If the resultant force of the object is zero, find the magnitude and direction angle of [math]F_3[/math].

(Calculator) Find the magnitude of the force required to keep a 400-kilogram container from sliding down a ramp inclined at [math]20^\circ[/math] from the horizontal.

(Calculator) Given two vectors [math]\overrightarrow{u}=5\textbf{i}+1\textbf{j}[/math] and [math]\overrightarrow{v}=2\textbf{i}-7\textbf{j}[/math], what is the direction angle of the vector [math]\overrightarrow{u}-\overrightarrow{v}[/math]?

Let [math]\overrightarrow{u}[/math] and [math]\overrightarrow{v}[/math] be two vectors. If [math]\overrightarrow{u}+\overrightarrow{v}[/math] and [math]\overrightarrow{u}-\overrightarrow{v}[/math] are parallel, then [math]\overrightarrow{u}[/math] and [math]\overrightarrow{v}[/math] are

An object is originally at [math](-1,3)[/math]. The object starts to move along the vector [math]-4\textbf{i}+3\textbf{j}[/math] for 7 seconds. If the speed of the object is 0.5 units per second, what is the new position of the object?