Problem $1$:

Two perpendicular vectors are given in terms of their components by

$F_1=F_{1x}i-F_{1y}j+2k$ and $F_2=i+3j+k$ with $F_{1y}=(F_{1x^{}}{)}^2+F_{1x}-1$

Use the dot product to determine the magnitude of the component, $F_{1x}.$

Problem $2$:

Suppose that the dot product of two vectors $U$ and $V$ is

$U*V=0.$ If $V0,$ what do you know about the vector $U?$

$U*V=0.$ If $V=0,$ what do you know about the vector $U?$

Problem $3$:

Find the angle $$ between the lines AB and AC as shown using the dot product of two vectors.

Problem $4$:

Cables extend from A to B and from A to C$.$ The cable AC exerts a $1000$ lb force $F$ at A $.$

What is the angle between the cables $AB$ and $AC?$

Determine the vector component of $F$ parallel to the cable $AB.$

Determine the vector component of $F$ perpendicular to the cable $AB.$

Hint: The vector component of $F$ parallel to the cable $AB$ is a vector.

$F_{\frac{\frac{?}{?}}{A}B}=(F_{\frac{\frac{?}{?}}{A}B})e_{AB},$ where $e_{AB}$ is a unit vector along $AB,F_{\frac{\frac{?}{?}}{A}B}$ can be determined using the

dot product of two vectors.