Displacement problem with unit vectors

[Ursa]

Homework Statement

In a meeting of mimes, mime 1 goes through a displacement d1 = (7.07 m)i + (3.94 m)j and and mime 2 goes through a displacement d2 = (-7.03 m)i + (4.1 m)j. What are (a)|d1 × d2|, (b)d1 · d2, (c)(d1 + d2) · d2, and (d) the component of d1 along the direction of d2?

Relevant Equations

a · b = ab *cos ϕ
a · b = ax*bx+ ay*by+az*bz
a = sqrt(ax^2+ay^2)
rx=ax+bx
rx=ax+bx

(a) I did (7.07*4.1)-(-7.03*3.94)=56.7 with this method I got this answer correct in my first attempt.

(b) This where I seem to have gone wrong. I used a · b = (a_xb_x +a_yb_y) then I used a = sqrt(ax²+ay²) to get a single number for the answer. Filling in the numbers 7.07*-7.03 + 4.94*4.1 = -49.7i+16.2j. Then sqrt(-49.²+16.2²)=52.3

(c) I first solved for (d1 + d2) with a_x+b_x 7.07+-7.03 + 4.94+4.1 = -0.04i+8.04j. Then used those unit vectors in the same way as in part (b) to get 32.97

(d) This one I didn't understand at all, I gave it a shot by trying to solve d₁ *cos ϕ. first trying to find ϕ by a · b = ab *cos ϕ
cos^-1((a · b)/ab). Finding d₁ and d₂ by a = sqrt(ax²+ay²
d₁=8.09
d₂= 8.14
cos^-1(52.3/8.09 * 8.14)=37.4

Then 8.14 cos 37.4= 6.43 mI just fundamentally don't understand vectors and really don't get how to do math with them. Any analyses of what I'm doing wrong, explanation and/or drawings would be appreciated.

 

[PeroK]

(b) This where I seem to have gone wrong. I used a · b = (a_xb_x +a_yb_y) then I used a = sqrt(ax²+ay²) to get a single number for the answer. Filling in the numbers 7.07*-7.03 + 4.94*4.1 = -49.7i+16.2j. Then sqrt(-49.²+16.2²)=52.3

The dot product is just a number so: $\vec d_1 \cdot \vec d_2 = 7.07*-7.03 + 3.94*4.1 = -49.7+16.2$

Note that should be $3.94$ in there.

(d) This one I didn't understand at all,

The component of one vector along another is what you get if you imagine taking the second vector as one of your basis vectors. Specifically, the component of $\vec d_1$ along $\vec d_2$ is defined as:
$$\frac{\vec d_1 \cdot \vec d_2}{|\vec d_2|}$$

 

[Ursa]

The dot product is just a number so: $\vec d_1 \cdot \vec d_2 = 7.07*-7.03 + 3.94*4.1 = -49.7+16.2$

So in order to get the answer I should just add $-49.7+16.2$ to get $-33.5$?

The component of one vector along another is what you get if you imagine taking the second vector as one of your basis vectors. Specifically, the component of $\vec d_1$ along $\vec d_2$ is defined as:
$$\frac{\vec d_1 \cdot \vec d_2}{|\vec d_2|}$$

Once I grasp how to get $\vec d_1 \cdot \vec d_2$ correct I have the top part, but could you expand on what exactly the denominator is? Will is be the $d_2$ = 8.14 which I calculated in my attempt?

 

[PeroK]

So in order to get the answer I should just add $-49.7+16.2$ to get $-33.5$?Once I grasp how to get $\vec d_1 \cdot \vec d_2$ correct I have the top part, but could you expand on what exactly the denominator is? Will is be the $d_2$ = 8.14 which I calculated in my attempt?

Yes, and yes. If $\vec d$ is a vector, then $|\vec d| \equiv d \equiv \sqrt{\vec d \cdot \vec d}$ where $|\vec d|$ and $d$ are both common notations for the magnitude of a vector.

 

[Ursa]

Yes, and yes. If $\vec d$ is a vector, then $|\vec d| \equiv d \equiv \sqrt{\vec d \cdot \vec d}$ where $|\vec d|$ and $d$ are both common notations for the magnitude of a vector.

Thank you, now I understand it a bit better.