Question

If the projection of $\mathrm{b}$ on a is twice the projection of a on b, then $|\mathrm{b}|-|\mathrm{a}|$ is equal to:
A) $|a-b|$
B) $\mathrm{a}|+| \mathrm{b}$
C) $|\text{b}|$$|a|$
D) $|a|$
E) 1

Answer 1

Hint:
The definition of scalar projection is the length of the vector projection. Recall that the dot product of a vector is a scalar quantity describing only the magnitude of a particular vector. A scalar projection is given by the dot product of a vector with a unit vector for that direction. Projection law states that in any triangle: Where $, \mathrm{A}, \mathrm{B}, \mathrm{C}$ are the three angles of the triangles and a, b, c are the corresponding opposite sides of the angles. Projection law or the formula of projection law express the algebraic sum of the projection of any two sides in terms of the third side.
The vector projection of a on b is a vector $\mathrm{a}_{1}$ which is either null or parallel to b. More exactly: ${ }^{*} \mathrm{a}_{1}=0$ if $\theta=90^{\circ},{ }^{*} \mathrm{a}_{1}$ and $\mathrm{b}$ have the same direction if $0 \leq \theta<90$ degrees, ${ }^{*}$$a_{1}$ and $\mathbf{b}$ have opposite directions if 90 degrees $<\theta \leq 180$ degrees.

Complete step by step solution:
In mathematics, a projection is a mapping of a set (or other mathematical structure) into a subset (or sub-structure), which is equal to its square for mapping composition (or, in other words, which is idempotent). An everyday example of a projection is the casting of shadows onto a plane (paper sheet).
Let projection of a on $\mathrm{b}=\mathrm{u}$
$\Rightarrow \mathrm{u}=|\mathrm{a}| \cos \theta=\dfrac{|\mathrm{a}|(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}})}{|\mathrm{a}||\mathrm{b}|}$
Let projection of $\mathrm{b}$ on $\mathrm{a}=\mathrm{v}$
$\Rightarrow \mathrm{v}=|\mathrm{b}| \cos \theta=\dfrac{|\mathrm{b}|(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}})}{|\mathrm{a}||\mathrm{b}|}$
Given, $\mathrm{v}=2 \mathrm{u}$
$\Rightarrow \dfrac{|\mathrm{a}|(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}})}{|\mathrm{a}||\mathrm{b}|}=2\left(\dfrac{|\mathrm{b}|(\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}})}{|\mathrm{a}||\mathrm{b}|}\right)$
$\Rightarrow|\mathrm{b}|=2|\mathrm{a}|$
$|\mathrm{b}|-|\mathrm{a}|=2|\mathrm{a}|-|\mathrm{a}|=|\mathrm{a}|$

Hence, the correct answer is option D.

Note:
The dot product of a with unit vector u denoted $\mathrm{a} \cdot \mathrm{u},$ is defined to be the projection of a in the direction of u or the amount that a is pointing in the same direction as unit vector u. The distance we travel in the direction of v, while traversing u is called the component of u with respect to v and is denoted comp vu. The vector parallel to v with magnitude comp vu, in the direction of v is called the projection of u onto v and is denoted proj vu.
The dot product of two Euclidean vectors a and b is defined by. where $\theta$ is the angle between a and b. In particular, if the vectors a and b are orthogonal (i.e., their angle $90^{\circ}$ ), then, which implies that. At the other extreme, if they are codirectional, then the angle between them is zero.