Let a = 5i - j - 3k and b = i + 3j + 5k be two vectors. Then which one of the following statements i

Q: Wait, why is the projection a scalar? I thought a projection was a vector.

That is a fantastic question! In vector algebra, we distinguish between the 'scalar projection' and the 'vector projection'. The scalar projection is a single number representing the signed length of the shadow. If you multiply this scalar by the unit vector of $\vec{b}$, you get the 'vector projection'. Here, the problem asks for the scalar value and the direction, which is effectively asking for the scalar projection and its orientation.

Q: What if the dot product was negative? Does that change the magnitude?

Great intuition! If the dot product $\vec{a} \cdot \vec{b}$ were negative, the scalar projection would also be negative. This doesn't change the 'length' in the absolute sense, but it tells us that the angle $\theta$ between the vectors is obtuse (greater than $90^\circ$). In that case, the projection would point in the direction opposite to $\vec{b}$. Since our dot product is positive, we know the angle is acute, so the direction is the same.

Q: Can I use the cross product to find the projection?

No, the cross product is designed to find a vector perpendicular to both $\vec{a}$ and $\vec{b}$, which is useful for calculating areas or normal vectors. For projections, the dot product is your best friend because it naturally contains the $\cos \theta$ term, which is exactly what we need to measure how much one vector aligns with another.

Question

Let $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$ be two vectors. Then which one of the following statements is TRUE?

Options:

(A) Projection of $\vec{a}$ on $\vec{b}$ is $\frac{17}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$.

(B) Projection of $\vec{a}$ on $\vec{b}$ is $\frac{17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$.

(C) Projection of $\vec{a}$ on $\vec{b}$ is $\frac{-17}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec{b}$.

(D) Projection of $\vec{a}$ on $\vec{b}$ is $\frac{-17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are going to peel back the layers of a fundamental concept in vector algebra: the projection. It is not just a formula; it is a way of understanding how one object influences another in 3D space.

Imagine you are standing in a room, and you have two vectors, $\vec{a} = 5\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$. You want to know how much of $\vec{a}$ is 'hidden' within $\vec{b}$. This is the essence of projection.

### The Flashlight Analogy

Think of vector $\vec{b}$ as a line on the floor. Now, imagine vector $\vec{a}$ is a stick floating in the air above it.

If you shine a flashlight directly down from above, perpendicular to the line of $\vec{b}$, the shadow cast by the stick onto that line is the projection. Mathematically, we are looking for the component of $\vec{a}$ that lies along the direction of $\vec{b}$.

The formula for this scalar projection is elegant and powerful:

$$	ext{Projection} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$$

### The Machinery of Calculation

To find this value, we need two pieces of information: the dot product of the two vectors and the magnitude of the vector we are projecting onto.

Let us start with the dot product, $\vec{a} \cdot \vec{b}$. We take the corresponding components of $\vec{a}$ and $\vec{b}$ and multiply them:

$$\vec{a} \cdot \vec{b} = (5)(1) + (-1)(3) + (3)(5)$$

Calculating this, we get $5 - 3 + 15$, which simplifies to $17$. This positive value is our numerator.

Next, we need the magnitude of $\vec{b}$, denoted as $|\vec{b}|$. This is the length of the vector, calculated using the Pythagorean theorem in 3D:

$$|\vec{b}| = \sqrt{1^2 + 3^2 + 5^2} = \sqrt{1 + 9 + 25} = \sqrt{35}$$

### The Final Synthesis

Now, we bring it all together. The scalar projection is simply the ratio of these two values:

$$	ext{Scalar Projection} = \frac{17}{\sqrt{35}}$$

But we are not done yet! The problem asks for the direction. Here is the secret: the dot product $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}| \cos 	heta$ tells us everything about the angle $	heta$ between the vectors.

Since our dot product is $17$, which is greater than $0$, we know that $\cos 	heta$ must be positive. This means the angle $	heta$ is acute (less than $90^\circ$). Therefore, the projection vector points in the same direction as $\vec{b}$.

We have successfully navigated the geometry of the problem. We found the magnitude of the shadow and determined its orientation. Keep practicing this visualization; it is the key to mastering vectors in **JEE Advanced**!

Animated Solution for Mathematics - Vector Algebra: Let $a = 5 \hat{i} - \hat{j} - 3 \hat{k}$ and $b = \hat{i} + 3 \hat{j} + 5 \hat{k}$ be two vectors. Then which one of the following statements is TRUE?

Select Answer:

Visualized Solution

Identify the Vectors $a$ and $b$

Concept of Scalar Projection

Formula for Projection

Raw Setup for Dot Product

Atomic Compute: Dot Product Components

Atomic Compute: Dot Product Result

Raw Setup for Magnitude of $b$

Atomic Compute: Magnitude Components

Atomic Compute: Magnitude Result

Final Scalar Projection Value

Determine the Direction

Conclusion and Final Answer

The Sigma Insight: Scalar (Dot) Product

Analyzing the Setup

The Flashlight Analogy

The Machinery of Calculation

The Final Synthesis

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Animated Solution for Mathematics - Vector Algebra: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a=5i^−j^​−3k^ and b=i^+3j^​+5k^ be two vectors. Then which one of the following statements is TRUE?