(a) In unit-vector notation, what is r⃗=a⃗-b+c⃗ if a⃗=5.0 î+4.0 ĵ-6.0 k̂, b⃗=-2.0 î+2.0 ĵ+3.0 k̂

step 1 For part A, we're given vectors A, vector B, and vector C. We're given values for those. And we

(a) In unit-vector notation, what is r⃗=a⃗-b+c⃗ if a⃗=5.0...

(a) In unit-vector notation, what is $\vec{r}=\vec{a}-b+\vec{c}$ if $\vec{a}=5.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}-6.0 \hat{\mathrm{k}}, \vec{b}=-2.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}},$ and $\vec{c}=4.0 \hat{\mathrm{i}}+$ $3.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}} ?(\mathrm{b})$ Calculate the angle between $\vec{r}$ and the positive $z$ axis. (c) What is the component of $\vec{a}$ along the direction of $\vec{b} ?$ (d) What is the component of $\vec{a}$ perpendicular to the direction of $\vec{b}$ but in the plane of $\vec{a}$ and $\vec{b}$ ?(Hint: For $(\mathrm{c}),$ see $\mathrm{Eq} \cdot 3-20$ and Fig. $3-18$ for $(\mathrm{d}),$ see $\mathrm{Eq} \cdot 3-24 . )$

(a) In unit-vector notation, what is $\vec{r}=\vec{a}-b+\vec{c}$ if
$\vec{a}=5.0 \hat{\mathrm{i}}+4.0 \hat{\mathrm{j}}-6.0 \hat{\mathrm{k}}, \vec{b}=-2.0 \hat{\mathrm{i}}+2.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}},$ and $\vec{c}=4.0 \hat{\mathrm{i}}+$
$3.0 \hat{\mathrm{j}}+2.0 \hat{\mathrm{k}} ?(\mathrm{b})$ Calculate the angle between $\vec{r}$ and the positive $z$ axis. (c) What is the component of $\vec{a}$ along the direction of $\vec{b} ?$ (d)
What is the component of $\vec{a}$ perpendicular to the direction of $\vec{b}$ but in the plane of $\vec{a}$ and $\vec{b}$ ?(Hint: For $(\mathrm{c}),$ see $\mathrm{Eq} \cdot 3-20$ and Fig. $3-18$ for $(\mathrm{d}),$ see $\mathrm{Eq} \cdot 3-24 . )$

Key Concepts

Vector Rejection

Vector rejection refers to the component of one vector that is perpendicular to another vector. It is found by subtracting the projected component (parallel to the given vector) from the original vector, thereby isolating the portion that lies in the direction orthogonal to the other vector. This concept is especially useful in decomposing vectors into parts that are parallel and perpendicular to a given direction.

Vector Projection

Vector projection is the operation of projecting one vector onto another, resulting in a component that is parallel to the second vector. This projection is used to find the part of a vector that lies in the direction of another vector, and is calculated using the dot product and the magnitude of the vector being projected onto.

Angle Between Vectors

The angle between two vectors can be determined by using the dot product formula, which relies on the magnitudes of the vectors and the cosine of the angle between them. This concept is important for understanding the directional relationship between vectors, such as the angle between a vector and an axis.

Dot Product

The dot product, or scalar product, of two vectors calculates a scalar quantity that includes the magnitude of both vectors and the cosine of the angle between them. It is used to determine the angle between vectors, to project one vector onto another, and to test for orthogonality between vectors.

Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors by adding or subtracting their respective components. These operations are fundamental for determining resultant vectors when multiple vectors are combined, as they allow one to compute the net effect or difference between vectors in any given direction.

Unit-Vector Notation

Unit-vector notation expresses a vector as a combination of its components along the coordinate axes, typically represented by i, j, and k in a three-dimensional space. This notation allows for straightforward addition, subtraction, and other vector operations by operating on the corresponding components.

Transcript

00:01 For part a, we're given vectors a, vector b, and vector c.

00:07 We're given values for those.

00:08 And we are asked to compute r, vector r when it's equal to vector a minus vector b plus vector c.

00:20 These are all vectors, of course.

00:23 Okay, well, recalling that you can only add and subtract like components, so i components, j -hack components, or k -hack components.

00:31 The first, the i -hack component is going to be five from vectors.

00:35 A minus minus 2 from vector b plus 4.

00:41 So 5 plus 2 plus 4, which is equal to 11.

00:45 So that's our ihat value, right? plus for j hat, we have 4 from vector a minus a 2 from vector b plus a 3 from vector c.

00:58 So 4 minus 2 plus 3 is equal to 5 in the j hat.

01:03 Right plus for the k hat we have minus six from vector a minus three from vector b plus two from vector c so that's a minus seven so this is minus seven in the k hat we can box that in as our solution to a okay okay now we're moving on to part b so um so we need to find the angle from the plus z axis by dotting or taking the scalar product of vector r with k hat.

01:40 Noting that that vector r has a magnitude, we can calculate this from, oh, excuse me, let's get rid of that, sorry about that.

01:54 We can calculate this from pythagorean theorem, so you take the square root of the, it's 11 squared plus 5 squared, plus minus 7 squared, well, a negative squared is a positive, so you can just write 7 squared.

02:11 So, carrying out this calculation, we find that this is equal to 14.

02:15 So r vector r dot k hat is equal to minus seven, since that's, since that's the k hat value from part a for vector r.

02:31 And also the dot product is equal to 14, the magnitude of vector r times one, which is the magnitude of vector k.

02:41 So anything times one is just itself.

02:42 So we don't need to write that.

02:44 So we can just write this as 14 actually, you don't need parentheses, and then multiplied by the cosine of the angle phi, which is what we're trying to find here.

02:54 Therefore, phi is equal to the inverse cosine of the ratio of negative 7 to 14 or negative 1 half.

03:06 Take the inverse cosine of negative 1 half and you find that this is equal to 120 degrees.

03:14 And we can box that in as our solution to b.

03:20 Okay...

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