The projections of a vector on the three coordinate axis are 6, -3, 2 respectively. The direction co

Q: Wait, why do we divide by the magnitude to get the direction cosines?

Great question! Think of it this way: a vector has both magnitude and direction. By dividing the vector by its own magnitude, we strip away the 'length' information, leaving behind only the 'direction' information. This resulting vector is a unit vector, and its components are precisely the cosines of the angles it makes with the axes—hence, direction cosines.

Q: Can direction cosines be negative?

Absolutely! A negative direction cosine simply means the vector makes an obtuse angle (greater than $90^{\circ}$) with that specific coordinate axis. In our case, the negative $y$-component tells us the vector points in the negative direction along the $y$-axis.

Q: Is there a quick way to check if my answer is correct?

Yes, there is a beautiful property! For any vector, the sum of the squares of its direction cosines must always equal one: $l^2 + m^2 + n^2 = 1$. If you square your results and add them up, you should get exactly $1$.

Question

The projections of a vector on the three coordinate axis are $6, -3, 2$ respectively. The direction cosines of the vector are :

Options:

(A) $\frac{6}{5}, \frac{-3}{5}, \frac{2}{5}$

(B) $\frac{6}{7}, \frac{-3}{7}, \frac{2}{7}$

(C) $\frac{-6}{7}, \frac{-3}{7}, \frac{2}{7}$

(D) $6, -3, 2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the origin of a 3D coordinate system. You are looking at a vector that stretches out into space, and you are told its projections on the $x$, $y$, and $z$ axes are $6$, $-3$, and $2$, respectively.

These projections are essentially the 'shadows' of the vector on each axis. By understanding these shadows, we can reconstruct the entire vector.

### Defining the Vector

Since we know the projections, we can immediately define our vector $\vec{r}$ as:

$$\vec{r} = 6\hat{i} - 3\hat{j} + 2\hat{k}$$

This vector starts at the origin and points directly to the coordinate $(6, -3, 2)$. It is a physical object in space, and our goal is to find its orientation—its direction cosines.

### Measuring the Reach: The Magnitude

Before we can find the direction, we need to know the total length of the vector. We use the distance formula in 3D space:

$$|\vec{r}| = \sqrt{x^2 + y^2 + z^2}$$

Substituting our values, we get:

$$|\vec{r}| = \sqrt{6^2 + (-3)^2 + 2^2}$$

Calculating the squares, we have $36 + 9 + 4$, which sums to $49$. The square root of $49$ is $7$. So, our vector has a total length of $7$ units.

### The DNA of Orientation: Direction Cosines

Now, we arrive at the heart of the problem. Direction cosines, denoted as $l$, $m$, and $n$, are the cosines of the angles the vector makes with the $x$, $y$, and $z$ axes. They are defined as:

$$l = \frac{x}{|\vec{r}|}, \quad m = \frac{y}{|\vec{r}|}, \quad n = \frac{z}{|\vec{r}|}$$

By dividing each component by the magnitude, we are essentially normalizing the vector.

For the $x$-axis:

$$l = \frac{6}{7}$$

For the $y$-axis:

$$m = \frac{-3}{7}$$

For the $z$-axis:

$$n = \frac{2}{7}$$

These three values, $(\frac{6}{7}, \frac{-3}{7}, \frac{2}{7})$, uniquely define the orientation of our vector in 3D space. It is elegant, precise, and perfectly describes how the vector is tilted relative to our coordinate system.

You have successfully mapped the vector's orientation! The final direction cosines are **$(\frac{6}{7}, -\frac{3}{7}, \frac{2}{7})$**.

Animated Solution for Mathematics - Vector Algebra: The projections of a vector on the three coordinate axis are $6, - 3, 2$ respectively. The direction cosines of the vector are :

Select Answer:

Visualized Solution

Visualizing the Projections

Defining the Vector $r$

Magnitude Formula

Raw Setup for Magnitude

Squaring the Components

Calculating Magnitude $∣ r ∣$

Direction Cosines Formula

Calculating $l$

Calculating $m$

Calculating $n$

Final Answer

The Sigma Insight: Components of a Vector

Analyzing the Setup

Defining the Vector

Measuring the Reach

The DNA of Orientation

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