# The points with position vectors $60\hat i+3\hat j,\:70\hat i-8\hat j\:and\:a\hat i-52\hat j$ are collinear if $a=?$

$\begin{array}{1 1} 40 \\ -40 \\ 80 \\ -80 \end{array}$

Let the points be $A(60\hat i+3\hat j),\:B(40\hat i-8\hat j) \:and\:C(a\hat i-52\hat j)$
$\overrightarrow {AB}=-20\hat i-11\hat j$
$\overrightarrow {BC}=(a-40)\hat i-44\hat j$
If $A,B,C$ are collinear, then $\overrightarrow {AB}=\lambda \overrightarrow {BC}$
for some constant $\lambda$
$\Rightarrow\:-80=a-40$
$\Rightarrow\:a=-40$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9755913019180298, "perplexity": 356.94354598654525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-22/segments/1526794867977.85/warc/CC-MAIN-20180527004958-20180527024958-00513.warc.gz"}