Q.Obtain the expressions for the rectangular (Cartesian) components of a vector in terms of (x), (y) and (z) axes.

Magnitude of A is 3, so the unit vector is (1/3)(sqrt5 i + 2 j). Answer (B).

Given A = sqrt5 i + 2 j.

Magnitude: |A| = sqrt((sqrt5)^2 + (2)^2) = sqrt(5 + 4) = sqrt(9) = 3.

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Magnitude = sqrt(Ax^2+Ay^2); direction theta = tan^-1(Ay/Ax). Answer (D).

For A = Ax i + Ay j:

Magnitude: |A| = sqrt(Ax^2 + Ay^2) (Pythagoras on the components).

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The unit vector along any vector $\vec{A}$ is $\hat{A} = \vec{A}/|\vec{A}|$; here $|\vec{A}| = \sqrt{6}$.

Given $\vec{A} = \hat{i} + 2\hat{j} + \hat{k}$, its magnitude is

$|\vec{A}| = \sqrt{1^2 + 2^2 + 1^2} = \sqrt{1+4+1} = \sqrt{6}$

The unit vector along $\vec{A}$ is then

$\hat{A} = \dfrac{\vec{A}}{|\vec{A}|} = \dfrac{\hat{i}+2\hat{j}+\hat{k}}{\sqrt{6}}$

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Dividing 3i + 4j by its magnitude 5 gives the unit vector 0.6 i + 0.8 j.

The magnitude of the vector A = 3i + 4j is |A| = sqrt(3^2 + 4^2) = sqrt(9+16) = sqrt(25) = 5.

A unit vector in the direction of A is given by A-hat = A / |A|. …

A vector's magnitude equals the square root of the sum of the squares of its perpendicular components: A = √(Ax² + Ay²).

A vector A→ in the x-y plane can be resolved into two mutually perpendicular components, Ax along the x-axis and Ay along the y-axis, so that A→ = Ax i^ + Ay j^. Geometrically, Ax and Ay form the two legs of a right triangle whose hypotenuse is A→ itself, so by the Pythagorean theorem:

A² = Ax² + Ay²

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The vector's components are (terminal point) minus (initial point), coordinate-wise.

Initial point $(2,1)$, terminal point $(-5,7)$.

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The dot product with $\hat i$ extracts the coefficient of $\hat i$, which is $3$.

Using $\hat i\cdot\hat i = 1$, $\hat j\cdot\hat i = 0$, $\hat k\cdot\hat i = 0$:

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For a vector A = 3i + 4j, the magnitude is sqrt(3^2 + 4^2) = 5.

For a vector written in component form as A = Ax i + Ay j, where i and j are unit vectors along the x and y axes, the magnitude is

|A| = sqrt(Ax^2 + Ay^2)

Here Ax = 3 and Ay = 4, so …

Position vector $=x\hat i+y\hat j+z\hat k$.

The position vector of a point $P(x,y,z)$ relative to the origin is $\vec{OP}=x\hat i+y\hat j+z\hat k$, with all compon …

The unit vector along A = 4i + 3j - 5k is A-hat = A/|A| ~ 0.566 i + 0.424 j - 0.707 k.

Step 1 - Magnitude of A: |A| = sqrt(A_x^2 + A_y^2 + A_z^2) = sqrt(4^2 + 3^2 + (-5)^2) = sqrt(16 + 9 + 25) = sqrt(50) = 5*sqrt(2) ~ 7.071.

Step 2 - Unit vector: A unit vector in the direction of A is defined as A-hat = A/|A| (a vector of magnitude 1 pointing the same way as A).

A-hat = (4i + 3j - 5k)/(5sqrt(2)) = (4/(5sqrt(2))) i + (3/(5sqrt(2))) j - (5/(5sqrt(2))) k

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