Latex:

The vector $\begin{pmatrix} k \\ 2 \end{pmatrix}$ is orthogonal to the vector $\begin{pmatrix} 3 \\ 5 \end{pmatrix}$. Find $k$.

Regular:
The vector <k, 2>, is orthogonal to the vector <3, 5>. Find k.

I can't seem to figure it out, I thought k would be 10/3

<k,2>•<3,5> = 0

3k+10 = 0
k = -10/3

gotta watch those ± signs!

Well, well, well, it seems like you're having some trouble with this math problem. Don't worry, I'll clown around and help you figure it out!

Let's recall that two vectors are orthogonal if their dot product is equal to zero. In other words, if you take the product of their corresponding components and add them up, the result should be zero.

So, let's do some clown math and calculate the dot product of these two vectors:

<k, 2> · <3, 5> = (k * 3) + (2 * 5) = 3k + 10

We want this dot product to be zero, so we set it equal to zero and solve for k:

3k + 10 = 0

Subtracting 10 from both sides:

3k = -10

Dividing by 3:

k = -10/3

And there you have it! The clown math tells us that k = -10/3. So, the vector <k, 2> is orthogonal to the vector <3, 5> when k is -10/3.

Keep up the good work, and remember, math can be a lot of fun, even when you're clowning around with it!

To find the value of k, we can use the fact that two vectors are orthogonal if their dot product is equal to zero.

The dot product of two vectors A = <a1, a2> and B = <b1, b2> is given by:

A · B = a1 * b1 + a2 * b2

In this case, we have the vector A = <k, 2> and the vector B = <3, 5>.

The dot product of A and B is:

A · B = (k * 3) + (2 * 5) = 3k + 10

Since the vectors A and B are orthogonal, their dot product must be zero. Therefore, we have:

3k + 10 = 0

To solve for k, we can subtract 10 from both sides:

3k = -10

Finally, we can divide both sides by 3:

k = -10/3

So, the value of k is -10/3.

To find the value of k, we can use the concept of vector orthogonality. Two vectors are orthogonal if their dot product is equal to zero. In other words, if the dot product of two vectors is zero, it means they are perpendicular to each other.

In this case, we have the two vectors: <k, 2> and <3, 5>. To check if they are orthogonal, we need to calculate their dot product.

The dot product of two vectors <a, b> and <c, d> is given by the formula: a*c + b*d.

So, for the given vectors, the dot product is:

(k * 3) + (2 * 5) = 3k + 10

Since these vectors are orthogonal, their dot product should be zero. Therefore, we can set up the equation:

3k + 10 = 0

To solve for k, we need to isolate k on one side of the equation:

3k = -10

Divide both sides of the equation by 3:

k = -10/3

So, the value of k is -10/3, not 10/3 as you initially thought.

Therefore, the vector <k, 2> is orthogonal to the vector <3, 5> when k is equal to -10/3.

Thanks!